Theorem ssiinf 3727

Description: Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)

Hypothesis

Ref	Expression
ssiinf.1	|- F/_ x C

Assertion

Ref	Expression
ssiinf	|- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B )

Proof of Theorem ssiinf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . . . 5 |- y e. _V
2		eliin 3683	. . . . 5 |- ( y e. _V -> ( y e. |^|_ x e. A B <-> A. x e. A y e. B ) )
3	1, 2	ax-mp 7	. . . 4 |- ( y e. |^|_ x e. A B <-> A. x e. A y e. B )
4	3	ralbii 2372	. . 3 |- ( A. y e. C y e. |^|_ x e. A B <-> A. y e. C A. x e. A y e. B )
5		ssiinf.1	. . . 4 |- F/_ x C
6		nfcv 2219	. . . 4 |- F/_ y A
7	5, 6	ralcomf 2515	. . 3 |- ( A. y e. C A. x e. A y e. B <-> A. x e. A A. y e. C y e. B )
8	4, 7	bitri 182	. 2 |- ( A. y e. C y e. |^|_ x e. A B <-> A. x e. A A. y e. C y e. B )
9		dfss3 2989	. 2 |- ( C C_ |^|_ x e. A B <-> A. y e. C y e. |^|_ x e. A B )
10		dfss3 2989	. . 3 |- ( C C_ B <-> A. y e. C y e. B )
11	10	ralbii 2372	. 2 |- ( A. x e. A C C_ B <-> A. x e. A A. y e. C y e. B )
12	8, 9, 11	3bitr4i 210	1 |- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B )

Syntax hints:   <-> wb 103   e. wcel 1433   F/_wnfc 2206   A.wral 2348   _Vcvv 2601   C_ wss 2973   |^|_ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-iin 3681

This theorem is referenced by:  ssiin  3728  dmiin  4598