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 2623	. . . 4 |- {y e. B | ph} = {y | (y e. B /\ ph)}
3	2	a1i 10	. . 3 |- (x e. A -> {y e. B | ph} = {y | (y e. B /\ ph)})
4	3	iuneq2i 3987	. 2 |- U_x e. A {y e. B | ph} = U_x e. A {y | (y e. B /\ ph)}
5		df-rab 2623	. . 3 |- {y e. B | E.x e. A ph} = {y | (y e. B /\ E.x e. A ph)}
6		r19.42v 2765	. . . 4 |- (E.x e. A (y e. B /\ ph) <-> (y e. B /\ E.x e. A ph))
7	6	abbii 2465	. . 3 |- {y | E.x e. A (y e. B /\ ph)} = {y | (y e. B /\ E.x e. A ph)}
8	5, 7	eqtr4i 2376	. 2 |- {y e. B | E.x e. A ph} = {y | E.x e. A (y e. B /\ ph)}
9	1, 4, 8	3eqtr4i 2383	1 |- U_x e. A {y e. B | ph} = {y e. B | E.x e. A ph}

Syntax hints:   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2615  {crab 2618  U_ciun 3969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iun 3971

This theorem is referenced by: (None)