Theorem iinss 3729

Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

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

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem iinss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . . 4 |- y e. _V
2		eliin 3683	. . . 4 |- ( y e. _V -> ( y e. |^|_ x e. A B <-> A. x e. A y e. B ) )
3	1, 2	ax-mp 7	. . 3 |- ( y e. |^|_ x e. A B <-> A. x e. A y e. B )
4		ssel 2993	. . . . 5 |- ( B C_ C -> ( y e. B -> y e. C ) )
5	4	reximi 2458	. . . 4 |- ( E. x e. A B C_ C -> E. x e. A ( y e. B -> y e. C ) )
6		r19.36av 2505	. . . 4 |- ( E. x e. A ( y e. B -> y e. C ) -> ( A. x e. A y e. B -> y e. C ) )
7	5, 6	syl 14	. . 3 |- ( E. x e. A B C_ C -> ( A. x e. A y e. B -> y e. C ) )
8	3, 7	syl5bi 150	. 2 |- ( E. x e. A B C_ C -> ( y e. |^|_ x e. A B -> y e. C ) )
9	8	ssrdv 3005	1 |- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1433   A.wral 2348   E.wrex 2349   _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-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-iin 3681

This theorem is referenced by:  riinm  3750  reliin  4477  cnviinm  4879  iinerm  6201