Theorem iunab 3300

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 3299	. 2 |- U_x e. A {y e. _V | ph} = {y e. _V | E.x e. A ph}
2		rabab 2308	. . . 4 |- {y e. _V | ph} = {y | ph}
3	2	a1i 8	. . 3 |- (x e. A -> {y e. _V | ph} = {y | ph})
4	3	iuneq2i 3276	. 2 |- U_x e. A {y e. _V | ph} = U_x e. A {y | ph}
5		rabab 2308	. 2 |- {y e. _V | E.x e. A ph} = {y | E.x e. A ph}
6	1, 4, 5	3eqtr3i 1918	1 |- U_x e. A {y | ph} = {y | E.x e. A ph}

Syntax hints:   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106  {crab 2108  _Vcvv 2292  U_ciun 3255

This theorem is referenced by:  iunid 3308  dfimafn2 4721  oarec 5244  iscst4 14522  sallnei 14873

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-in 2603  df-ss 2605  df-iun 3257

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