▁is ▁. ▁ ▁The ▁ax iom ▁of ▁union ▁states ▁that ▁for ▁any ▁set ▁of ▁sets ▁ ▁there ▁is ▁a ▁set ▁ ▁containing ▁every ▁element ▁that ▁is ▁a ▁member ▁of ▁some ▁member ▁of ▁: ▁▁ ▁Although ▁this ▁formula ▁doesn ' t ▁directly ▁assert ▁the ▁existence ▁of ▁, ▁the ▁set ▁ ▁can ▁be ▁constructed ▁from ▁ ▁in ▁the ▁above ▁using ▁the ▁ax iom ▁schema ▁of ▁specification : ▁▁ 6 . ▁A xi om ▁schema ▁of ▁replacement ▁▁ ▁The ▁ax iom ▁schema ▁of ▁replacement ▁as ser ts ▁that ▁the ▁image ▁of ▁a ▁set ▁under ▁any ▁defin able ▁function ▁will ▁also ▁fall ▁inside ▁a ▁set . ▁ ▁Form ally , ▁let ▁ ▁be ▁any ▁formula ▁in ▁the ▁language ▁of ▁Z FC ▁whose ▁free ▁variables ▁are ▁among ▁, ▁so ▁that ▁in ▁particular ▁ ▁is ▁not ▁free ▁in ▁. ▁Then : ▁▁▁▁ ▁In ▁other ▁words , ▁if ▁the ▁relation ▁ ▁represents ▁a ▁defin able ▁function ▁, ▁ ▁represents ▁its ▁domain , ▁and ▁ ▁is ▁a ▁set ▁for ▁every ▁, ▁then ▁the ▁range ▁of ▁ ▁is ▁a ▁subset ▁of ▁some ▁set ▁. ▁The ▁form ▁stated ▁here , ▁in ▁which ▁ ▁may ▁be ▁larger ▁than ▁strictly ▁necessary , ▁is ▁sometimes ▁called ▁the ▁ax iom ▁schema ▁of ▁collection . ▁▁ 7 . ▁A xi om ▁of ▁infinity ▁▁ ▁Let ▁ ▁abbre vi ate ▁, ▁where ▁ ▁is ▁some ▁set . ▁( We ▁can ▁see ▁that ▁ ▁is ▁a ▁valid ▁set ▁by ▁applying ▁the ▁A xi om ▁of ▁P air ing ▁with ▁ ▁so ▁that ▁the ▁set ▁ ▁is ▁). ▁Then ▁there ▁exists