Theorem iunrab 4013

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 4012	. 2 |- U_ x e. A { y | ( y e. B /\ ph ) } = { y | E. x e. A ( y e. B /\ ph ) }
2		df-rab 2517	. . . 4 |- { y e. B | ph } = { y | ( y e. B /\ ph ) }
3	2	a1i 9	. . 3 |- ( x e. A -> { y e. B | ph } = { y | ( y e. B /\ ph ) } )
4	3	iuneq2i 3983	. 2 |- U_ x e. A { y e. B | ph } = U_ x e. A { y | ( y e. B /\ ph ) }
5		df-rab 2517	. . 3 |- { y e. B | E. x e. A ph } = { y | ( y e. B /\ E. x e. A ph ) }
6		r19.42v 2688	. . . 4 |- ( E. x e. A ( y e. B /\ ph ) <-> ( y e. B /\ E. x e. A ph ) )
7	6	abbii 2345	. . 3 |- { y | E. x e. A ( y e. B /\ ph ) } = { y | ( y e. B /\ E. x e. A ph ) }
8	5, 7	eqtr4i 2253	. 2 |- { y e. B | E. x e. A ph } = { y | E. x e. A ( y e. B /\ ph ) }
9	1, 4, 8	3eqtr4i 2260	1 |- U_ x e. A { y e. B | ph } = { y e. B | E. x e. A ph }

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   {cab 2215   E.wrex 2509   {crab 2512   U_ciun 3965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-in 3203  df-ss 3210  df-iun 3967

This theorem is referenced by:  hashrabrex  11992  phisum  12763  lgsquadlem1  15756  lgsquadlem2  15757