Theorem iunrab 2022

Description: The indexed union of a restricted class abstraction.

Assertion

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

Distinct variable group(s):   y,A   x,y,B

Proof of Theorem iunrab

Step	Hyp	Ref	Expression
1		df-rab 1208	. 2 |- {z e. B | E.x e. A [z / y]ph} = {z | (z e. B /\ E.x e. A [z / y]ph)}
2		ax-17 925	. . 3 |- (x e. B -> A.y x e. B)
3		ax-17 925	. . 3 |- (x e. B -> A.z x e. B)
4		ax-17 925	. . 3 |- (E.x e. A ph -> A.zE.x e. A ph)
5		ax-17 925	. . . 4 |- (x e. A -> A.y x e. A)
6		hbs1 986	. . . 4 |- ([z / y]ph -> A.y[z / y]ph)
7	5, 6	hbrex 1238	. . 3 |- (E.x e. A [z / y]ph -> A.yE.x e. A [z / y]ph)
8		sbequ12 865	. . . 4 |- (y = z -> (ph <-> [z / y]ph))
9	8	birexdv 1220	. . 3 |- (y = z -> (E.x e. A ph <-> E.x e. A [z / y]ph))
10	2, 3, 4, 7, 9	cbvrab 1425	. 2 |- {y e. B | E.x e. A ph} = {z e. B | E.x e. A [z / y]ph}
11		eliun 1998	. . . 4 |- (z e. U.x e. A {y e. B | ph} <-> E.x e. A z e. {y e. B | ph})
12	2	elrabsf 1456	. . . . 5 |- (z e. {y e. B | ph} <-> (z e. B /\ [z / y]ph))
13	12	birex 1224	. . . 4 |- (E.x e. A z e. {y e. B | ph} <-> E.x e. A (z e. B /\ [z / y]ph))
14		r19.42v 1303	. . . 4 |- (E.x e. A (z e. B /\ [z / y]ph) <-> (z e. B /\ E.x e. A [z / y]ph))
15	11, 13, 14	3bitr 155	. . 3 |- (z e. U.x e. A {y e. B | ph} <-> (z e. B /\ E.x e. A [z / y]ph))
16	15	biabri 1180	. 2 |- U.x e. A {y e. B | ph} = {z | (z e. B /\ E.x e. A [z / y]ph)}
17	1, 10, 16	3eqtr4r 1127	1 |- U.x e. A {y e. B | ph} = {y e. B | E.x e. A ph}

Syntax hints:   /\ wa 196   = weq 797  [wsb 852  {cab 1090   = wceq 1091   e. wcel 1092  E.wrex 1202  {crab 1204  U.ciun 1994

This theorem is referenced by:  iunab 2023

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-iun 1996

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