Theorem iinss 3979

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 2775	. . . 4 |- y e. _V
2		eliin 3932	. . . 4 |- ( y e. _V -> ( y e. |^|_ x e. A B <-> A. x e. A y e. B ) )
3	1, 2	ax-mp 5	. . 3 |- ( y e. |^|_ x e. A B <-> A. x e. A y e. B )
4		ssel 3187	. . . . 5 |- ( B C_ C -> ( y e. B -> y e. C ) )
5	4	reximi 2603	. . . 4 |- ( E. x e. A B C_ C -> E. x e. A ( y e. B -> y e. C ) )
6		r19.36av 2657	. . . 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	biimtrid 152	. 2 |- ( E. x e. A B C_ C -> ( y e. |^|_ x e. A B -> y e. C ) )
9	8	ssrdv 3199	1 |- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2176   A.wral 2484   E.wrex 2485   _Vcvv 2772   C_ wss 3166   |^|_ciin 3928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-iin 3930

This theorem is referenced by:  riinm  4000  reliin  4797  cnviinm  5224  iinerm  6694