Theorem iinrabm 4028

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 3584	. . 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 2347	. 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 2517	. . . . 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 3984	. . 3 |- |^|_ x e. A { y e. B | ph } = |^|_ x e. A { y | ( y e. B /\ ph ) }
6		iinab 4027	. . 3 |- |^|_ x e. A { y | ( y e. B /\ ph ) } = { y | A. x e. A ( y e. B /\ ph ) }
7	5, 6	eqtri 2250	. 2 |- |^|_ x e. A { y e. B | ph } = { y | A. x e. A ( y e. B /\ ph ) }
8		df-rab 2517	. 2 |- { y e. B | A. x e. A ph } = { y | ( y e. B /\ A. x e. A ph ) }
9	2, 7, 8	3eqtr4g 2287	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 104   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   A.wral 2508   {crab 2512   |^|_ciin 3966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-v 2801  df-iin 3968

This theorem is referenced by: (None)