Theorem iinrabm 3928

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 3501	. . 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 2284	. 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 2453	. . . . 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 3885	. . 3 |- |^|_ x e. A { y e. B | ph } = |^|_ x e. A { y | ( y e. B /\ ph ) }
6		iinab 3927	. . 3 |- |^|_ x e. A { y | ( y e. B /\ ph ) } = { y | A. x e. A ( y e. B /\ ph ) }
7	5, 6	eqtri 2186	. 2 |- |^|_ x e. A { y e. B | ph } = { y | A. x e. A ( y e. B /\ ph ) }
8		df-rab 2453	. 2 |- { y e. B | A. x e. A ph } = { y | ( y e. B /\ A. x e. A ph ) }
9	2, 7, 8	3eqtr4g 2224	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 1343   E.wex 1480   e. wcel 2136   {cab 2151   A.wral 2444   {crab 2448   |^|_ciin 3867

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728  df-iin 3869

This theorem is referenced by: (None)