Theorem iunrab 3933

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 3932	. 2 |- U_ x e. A { y | ( y e. B /\ ph ) } = { y | E. x e. A ( y e. B /\ ph ) }
2		df-rab 2464	. . . 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 3904	. 2 |- U_ x e. A { y e. B | ph } = U_ x e. A { y | ( y e. B /\ ph ) }
5		df-rab 2464	. . 3 |- { y e. B | E. x e. A ph } = { y | ( y e. B /\ E. x e. A ph ) }
6		r19.42v 2634	. . . 4 |- ( E. x e. A ( y e. B /\ ph ) <-> ( y e. B /\ E. x e. A ph ) )
7	6	abbii 2293	. . 3 |- { y | E. x e. A ( y e. B /\ ph ) } = { y | ( y e. B /\ E. x e. A ph ) }
8	5, 7	eqtr4i 2201	. 2 |- { y e. B | E. x e. A ph } = { y | E. x e. A ( y e. B /\ ph ) }
9	1, 4, 8	3eqtr4i 2208	1 |- U_ x e. A { y e. B | ph } = { y e. B | E. x e. A ph }

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2148   {cab 2163   E.wrex 2456   {crab 2459   U_ciun 3886

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-in 3135  df-ss 3142  df-iun 3888

This theorem is referenced by:  hashrabrex  11481  phisum  12231