Theorem iunrab 2596

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 groups:   y,A   x,y,B

Proof of Theorem iunrab

Step	Hyp	Ref	Expression
1		df-rab 1652	. 2 |- {z e. B | E.x e. A [z / y]ph} = {z | (z e. B /\ E.x e. A [z / y]ph)}
2		ax-17 971	. . 3 |- (x e. B -> A.y x e. B)
3		ax-17 971	. . 3 |- (x e. B -> A.z x e. B)
4		ax-17 971	. . 3 |- (E.x e. A ph -> A.zE.x e. A ph)
5		ax-17 971	. . . 4 |- (x e. A -> A.y x e. A)
6		hbs1 1332	. . . 4 |- ([z / y]ph -> A.y[z / y]ph)
7	5, 6	hbrex 1688	. . 3 |- (E.x e. A [z / y]ph -> A.yE.x e. A [z / y]ph)
8		sbequ12 1181	. . . 4 |- (y = z -> (ph <-> [z / y]ph))
9	8	rexbidv 1664	. . 3 |- (y = z -> (E.x e. A ph <-> E.x e. A [z / y]ph))
10	2, 3, 4, 7, 9	cbvrab 1910	. 2 |- {y e. B | E.x e. A ph} = {z e. B | E.x e. A [z / y]ph}
11		eliun 2570	. . . 4 |- (z e. U_x e. A {y e. B | ph} <-> E.x e. A z e. {y e. B | ph})
12	2	elrabsf 1963	. . . . 5 |- (z e. {y e. B | ph} <-> (z e. B /\ [z / y]ph))
13	12	rexbii 1668	. . . 4 |- (E.x e. A z e. {y e. B | ph} <-> E.x e. A (z e. B /\ [z / y]ph))
14		r19.42v 1764	. . . 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 177	. . 3 |- (z e. U_x e. A {y e. B | ph} <-> (z e. B /\ E.x e. A [z / y]ph))
16	15	abbi2i 1574	. 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 1506	1 |- U_x e. A {y e. B | ph} = {y e. B | E.x e. A ph}

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  [wsbc 1170  {cab 1463  E.wrex 1646  {crab 1648  U_ciun 2566

This theorem is referenced by:  iunab 2597

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-iun 2568

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