Theorem riinrab 4596

Description: Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

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

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

Allowed substitution hints:   ph( x, y)

Proof of Theorem riinrab

Step	Hyp	Ref	Expression
1		riin0 4594	. . 3 |- ( X = (/) -> ( A i^i |^|_ x e. X { y e. A | ph } ) = A )
2		rzal 4073	. . . . 5 |- ( X = (/) -> A. x e. X ph )
3	2	ralrimivw 2967	. . . 4 |- ( X = (/) -> A. y e. A A. x e. X ph )
4		rabid2 3118	. . . 4 |- ( A = { y e. A | A. x e. X ph } <-> A. y e. A A. x e. X ph )
5	3, 4	sylibr 224	. . 3 |- ( X = (/) -> A = { y e. A | A. x e. X ph } )
6	1, 5	eqtrd 2656	. 2 |- ( X = (/) -> ( A i^i |^|_ x e. X { y e. A | ph } ) = { y e. A | A. x e. X ph } )
7		ssrab2 3687	. . . . 5 |- { y e. A | ph } C_ A
8	7	rgenw 2924	. . . 4 |- A. x e. X { y e. A | ph } C_ A
9		riinn0 4595	. . . 4 |- ( ( A. x e. X { y e. A | ph } C_ A /\ X =/= (/) ) -> ( A i^i |^|_ x e. X { y e. A | ph } ) = |^|_ x e. X { y e. A | ph } )
10	8, 9	mpan 706	. . 3 |- ( X =/= (/) -> ( A i^i |^|_ x e. X { y e. A | ph } ) = |^|_ x e. X { y e. A | ph } )
11		iinrab 4582	. . 3 |- ( X =/= (/) -> |^|_ x e. X { y e. A | ph } = { y e. A | A. x e. X ph } )
12	10, 11	eqtrd 2656	. 2 |- ( X =/= (/) -> ( A i^i |^|_ x e. X { y e. A | ph } ) = { y e. A | A. x e. X ph } )
13	6, 12	pm2.61ine 2877	1 |- ( A i^i |^|_ x e. X { y e. A | ph } ) = { y e. A | A. x e. X ph }

Syntax hints:   = wceq 1483   =/= wne 2794   A.wral 2912   {crab 2916   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-rex 2918  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:  acsfn1  16322  acsfn1c  16323  acsfn2  16324  cntziinsn  17767  csscld  23048  acsfn1p  37769