Theorem iinrabm 3719

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 3314	. . 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 2155	. 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 2315	. . . . 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 3676	. . 3 |- |^|_ x e. A { y e. B | ph } = |^|_ x e. A { y | ( y e. B /\ ph ) }
6		iinab 3718	. . 3 |- |^|_ x e. A { y | ( y e. B /\ ph ) } = { y | A. x e. A ( y e. B /\ ph ) }
7	5, 6	eqtri 2060	. 2 |- |^|_ x e. A { y e. B | ph } = { y | A. x e. A ( y e. B /\ ph ) }
8		df-rab 2315	. 2 |- { y e. B | A. x e. A ph } = { y | ( y e. B /\ A. x e. A ph ) }
9	2, 7, 8	3eqtr4g 2097	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 97   = wceq 1243   E.wex 1381   e. wcel 1393   {cab 2026   A.wral 2306   {crab 2310   |^|_ciin 3658

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-iin 3660

This theorem is referenced by: (None)