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Theorem pthsfval 26520
 Description: The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)
Assertion
Ref Expression
pthsfval (Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}
Distinct variable group:   𝑓,𝐺,𝑝

Proof of Theorem pthsfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 biidd 252 . . . 4 ((⊤ ∧ 𝑔 = 𝐺) → ((Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅) ↔ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)))
2 wksv 26419 . . . . . 6 {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
3 trliswlk 26497 . . . . . . 7 (𝑓(Trails‘𝐺)𝑝𝑓(Walks‘𝐺)𝑝)
43ssopab2i 4973 . . . . . 6 {⟨𝑓, 𝑝⟩ ∣ 𝑓(Trails‘𝐺)𝑝} ⊆ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝}
52, 4ssexi 4773 . . . . 5 {⟨𝑓, 𝑝⟩ ∣ 𝑓(Trails‘𝐺)𝑝} ∈ V
65a1i 11 . . . 4 (⊤ → {⟨𝑓, 𝑝⟩ ∣ 𝑓(Trails‘𝐺)𝑝} ∈ V)
7 df-pths 26515 . . . . 5 Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
8 3anass 1040 . . . . . . 7 ((𝑓(Trails‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅) ↔ (𝑓(Trails‘𝑔)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)))
98opabbii 4689 . . . . . 6 {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅))}
109mpteq2i 4711 . . . . 5 (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}) = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅))})
117, 10eqtri 2643 . . . 4 Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅))})
121, 6, 11fvmptopab 6662 . . 3 (⊤ → (Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅))})
1312trud 1490 . 2 (Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅))}
14 3anass 1040 . . . 4 ((𝑓(Trails‘𝐺)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅) ↔ (𝑓(Trails‘𝐺)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)))
1514bicomi 214 . . 3 ((𝑓(Trails‘𝐺)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)) ↔ (𝑓(Trails‘𝐺)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅))
1615opabbii 4689 . 2 {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅))} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}
1713, 16eqtri 2643 1 (Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ⊤wtru 1481   ∈ wcel 1987  Vcvv 3190   ∩ cin 3559  ∅c0 3897  {cpr 4157   class class class wbr 4623  {copab 4682   ↦ cmpt 4683  ◡ccnv 5083   ↾ cres 5086   “ cima 5087  Fun wfun 5851  ‘cfv 5857  (class class class)co 6615  0cc0 9896  1c1 9897  ..^cfzo 12422  #chash 13073  Walkscwlks 26396  Trailsctrls 26490  Pathscpths 26511 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-hash 13074  df-word 13254  df-wlks 26399  df-trls 26492  df-pths 26515 This theorem is referenced by:  ispth  26522
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