Theorem iinrabm 3822

Description: Indexed intersection of a restricted class builder. (Contributed by Jim Kingdon, 16-Aug-2018.)

Assertion

Ref	Expression
iinrabm	|- ( E. x x e. A -> |^|_ x e. A { y e. B | ph } = { y e. B | A. x e. A ph } )

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

Allowed substitution hints:   ph( x, y)   B( y)

Proof of Theorem iinrabm

Step	Hyp	Ref	Expression
1		r19.28mv 3402	. . 3 |- ( E. x x e. A -> ( A. x e. A ( y e. B /\ ph ) <-> ( y e. B /\ A. x e. A ph ) ) )
2	1	abbidv 2217	. 2 |- ( E. x x e. A -> { y | A. x e. A ( y e. B /\ ph ) } = { y | ( y e. B /\ A. x e. A ph ) } )
3		df-rab 2384	. . . . 5 |- { y e. B | ph } = { y | ( y e. B /\ ph ) }
4	3	a1i 9	. . . 4 |- ( x e. A -> { y e. B | ph } = { y | ( y e. B /\ ph ) } )
5	4	iineq2i 3779	. . 3 |- |^|_ x e. A { y e. B | ph } = |^|_ x e. A { y | ( y e. B /\ ph ) }
6		iinab 3821	. . 3 |- |^|_ x e. A { y | ( y e. B /\ ph ) } = { y | A. x e. A ( y e. B /\ ph ) }
7	5, 6	eqtri 2120	. 2 |- |^|_ x e. A { y e. B | ph } = { y | A. x e. A ( y e. B /\ ph ) }
8		df-rab 2384	. 2 |- { y e. B | A. x e. A ph } = { y | ( y e. B /\ A. x e. A ph ) }
9	2, 7, 8	3eqtr4g 2157	1 |- ( E. x x e. A -> |^|_ x e. A { y e. B | ph } = { y e. B | A. x e. A ph } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1299   E.wex 1436   e. wcel 1448   {cab 2086   A.wral 2375   {crab 2379   |^|_ciin 3761

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rab 2384  df-v 2643  df-iin 3763

This theorem is referenced by: (None)