Theorem iunrab 4567

Description: The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
iunrab	|- U_ x e. A { y e. B | ph } = { y e. B | E. x e. A ph }

Distinct variable groups:   y, A   x, y   x, B

Allowed substitution hints:   ph( x, y)   A( x)   B( y)

Proof of Theorem iunrab

Step	Hyp	Ref	Expression
1		iunab 4566	. 2 |- U_ x e. A { y | ( y e. B /\ ph ) } = { y | E. x e. A ( y e. B /\ ph ) }
2		df-rab 2921	. . . 4 |- { y e. B | ph } = { y | ( y e. B /\ ph ) }
3	2	a1i 11	. . 3 |- ( x e. A -> { y e. B | ph } = { y | ( y e. B /\ ph ) } )
4	3	iuneq2i 4539	. 2 |- U_ x e. A { y e. B | ph } = U_ x e. A { y | ( y e. B /\ ph ) }
5		df-rab 2921	. . 3 |- { y e. B | E. x e. A ph } = { y | ( y e. B /\ E. x e. A ph ) }
6		r19.42v 3092	. . . 4 |- ( E. x e. A ( y e. B /\ ph ) <-> ( y e. B /\ E. x e. A ph ) )
7	6	abbii 2739	. . 3 |- { y | E. x e. A ( y e. B /\ ph ) } = { y | ( y e. B /\ E. x e. A ph ) }
8	5, 7	eqtr4i 2647	. 2 |- { y e. B | E. x e. A ph } = { y | E. x e. A ( y e. B /\ ph ) }
9	1, 4, 8	3eqtr4i 2654	1 |- U_ x e. A { y e. B | ph } = { y e. B | E. x e. A ph }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   {crab 2916   U_ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-iun 4522

This theorem is referenced by:  hashrabrex  14557  incexc2  14570  phisum  15495  itg2monolem1  23517  aannenlem1  24083  musum  24917  lgsquadlem1  25105  lgsquadlem2  25106  edglnl  26038  iunpreima  29383  poimirlem27  33436  cnambfre  33458  mapdval3N  36920  mapdval5N  36922  fiphp3d  37383