Theorem iinrab2 4583

Description: Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)

Assertion

Ref	Expression
iinrab2	|- ( |^|_ x e. A { y e. B | ph } i^i B ) = { y e. B | A. x e. A ph }

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

Allowed substitution hints:   ph( x, y)

Proof of Theorem iinrab2

Step	Hyp	Ref	Expression
1		iineq1 4535	. . . . . 6 |- ( A = (/) -> |^|_ x e. A { y e. B | ph } = |^|_ x e. (/) { y e. B | ph } )
2		0iin 4578	. . . . . 6 |- |^|_ x e. (/) { y e. B | ph } = _V
3	1, 2	syl6eq 2672	. . . . 5 |- ( A = (/) -> |^|_ x e. A { y e. B | ph } = _V )
4	3	ineq1d 3813	. . . 4 |- ( A = (/) -> ( |^|_ x e. A { y e. B | ph } i^i B ) = ( _V i^i B ) )
5		incom 3805	. . . . 5 |- ( _V i^i B ) = ( B i^i _V )
6		inv1 3970	. . . . 5 |- ( B i^i _V ) = B
7	5, 6	eqtri 2644	. . . 4 |- ( _V i^i B ) = B
8	4, 7	syl6eq 2672	. . 3 |- ( A = (/) -> ( |^|_ x e. A { y e. B | ph } i^i B ) = B )
9		rzal 4073	. . . 4 |- ( A = (/) -> A. x e. A A. y e. B ph )
10		rabid2 3118	. . . . 5 |- ( B = { y e. B | A. x e. A ph } <-> A. y e. B A. x e. A ph )
11		ralcom 3098	. . . . 5 |- ( A. y e. B A. x e. A ph <-> A. x e. A A. y e. B ph )
12	10, 11	bitr2i 265	. . . 4 |- ( A. x e. A A. y e. B ph <-> B = { y e. B | A. x e. A ph } )
13	9, 12	sylib 208	. . 3 |- ( A = (/) -> B = { y e. B | A. x e. A ph } )
14	8, 13	eqtrd 2656	. 2 |- ( A = (/) -> ( |^|_ x e. A { y e. B | ph } i^i B ) = { y e. B | A. x e. A ph } )
15		iinrab 4582	. . . 4 |- ( A =/= (/) -> |^|_ x e. A { y e. B | ph } = { y e. B | A. x e. A ph } )
16	15	ineq1d 3813	. . 3 |- ( A =/= (/) -> ( |^|_ x e. A { y e. B | ph } i^i B ) = ( { y e. B | A. x e. A ph } i^i B ) )
17		ssrab2 3687	. . . 4 |- { y e. B | A. x e. A ph } C_ B
18		dfss 3589	. . . 4 |- ( { y e. B | A. x e. A ph } C_ B <-> { y e. B | A. x e. A ph } = ( { y e. B | A. x e. A ph } i^i B ) )
19	17, 18	mpbi 220	. . 3 |- { y e. B | A. x e. A ph } = ( { y e. B | A. x e. A ph } i^i B )
20	16, 19	syl6eqr 2674	. 2 |- ( A =/= (/) -> ( |^|_ x e. A { y e. B | ph } i^i B ) = { y e. B | A. x e. A ph } )
21	14, 20	pm2.61ine 2877	1 |- ( |^|_ x e. A { y e. B | ph } i^i B ) = { y e. B | A. x e. A ph }

Syntax hints:   = wceq 1483   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-iin 4523

This theorem is referenced by: (None)