Theorem iinrab 3318

Description: Indexed intersection of a restricted class builder.

Assertion

Ref	Expression
iinrab	|- (A =/= (/) -> |^|_x e. A {y e. B | ph} = {y e. B | A.x e. A ph})

Distinct variable groups:   y,A,x   x,B

Proof of Theorem iinrab

Step	Hyp	Ref	Expression
1		r19.28zv 2964	. . 3 |- (A =/= (/) -> (A.x e. A (y e. B /\ ph) <-> (y e. B /\ A.x e. A ph)))
2	1	abbidv 2008	. 2 |- (A =/= (/) -> {y | A.x e. A (y e. B /\ ph)} = {y | (y e. B /\ A.x e. A ph)})
3		df-rab 2112	. . . . 5 |- {y e. B | ph} = {y | (y e. B /\ ph)}
4	3	a1i 8	. . . 4 |- (x e. A -> {y e. B | ph} = {y | (y e. B /\ ph)})
5	4	iineq2i 3277	. . 3 |- |^|_x e. A {y e. B | ph} = |^|_x e. A {y | (y e. B /\ ph)}
6		iinab 3317	. . 3 |- |^|_x e. A {y | (y e. B /\ ph)} = {y | A.x e. A (y e. B /\ ph)}
7	5, 6	eqtri 1908	. 2 |- |^|_x e. A {y e. B | ph} = {y | A.x e. A (y e. B /\ ph)}
8		df-rab 2112	. 2 |- {y e. B | A.x e. A ph} = {y | (y e. B /\ A.x e. A ph)}
9	2, 7, 8	3eqtr4g 1953	1 |- (A =/= (/) -> |^|_x e. A {y e. B | ph} = {y e. B | A.x e. A ph})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017  A.wral 2105  {crab 2108  (/)c0 2875  |^|_ciin 3256

This theorem is referenced by:  iinrab2 3319  pmapglbx 17251

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-ne 2019  df-ral 2109  df-rab 2112  df-v 2294  df-dif 2597  df-nul 2876  df-iin 3258

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