Theorem iunab 2597

Description: The indexed union of a class abstraction.

Assertion

Ref	Expression
iunab	|- U_x e. A {y | ph} = {y | E.x e. A ph}

Distinct variable groups:   y,A   x,y

Proof of Theorem iunab

Step	Hyp	Ref	Expression
1		iunrab 2596	. 2 |- U_x e. A {y e. V | ph} = {y e. V | E.x e. A ph}
2		rabab 1822	. . . 4 |- {y e. V | ph} = {y | ph}
3	2	a1i 8	. . 3 |- (x e. A -> {y e. V | ph} = {y | ph})
4	3	iuneq2i 2580	. 2 |- U_x e. A {y e. V | ph} = U_x e. A {y | ph}
5		rabab 1822	. 2 |- {y e. V | E.x e. A ph} = {y | E.x e. A ph}
6	1, 4, 5	3eqtr3 1503	1 |- U_x e. A {y | ph} = {y | E.x e. A ph}

Syntax hints:   = wceq 956   e. wcel 958  {cab 1463  E.wrex 1646  {crab 1648  Vcvv 1811  U_ciun 2566

This theorem is referenced by:  dfimafn2 3762  oarec 4196

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-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-in 2051  df-ss 2053  df-iun 2568

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