Theorem iinrabm 3870

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 3450	. . 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 2255	. 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 2423	. . . . 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 3827	. . 3 |- |^|_ x e. A { y e. B | ph } = |^|_ x e. A { y | ( y e. B /\ ph ) }
6		iinab 3869	. . 3 |- |^|_ x e. A { y | ( y e. B /\ ph ) } = { y | A. x e. A ( y e. B /\ ph ) }
7	5, 6	eqtri 2158	. 2 |- |^|_ x e. A { y e. B | ph } = { y | A. x e. A ( y e. B /\ ph ) }
8		df-rab 2423	. 2 |- { y e. B | A. x e. A ph } = { y | ( y e. B /\ A. x e. A ph ) }
9	2, 7, 8	3eqtr4g 2195	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 1331   E.wex 1468   e. wcel 1480   {cab 2123   A.wral 2414   {crab 2418   |^|_ciin 3809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683  df-iin 3811

This theorem is referenced by: (None)