Theorem iinss 3940

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 2742	. . . 4 |- y e. _V
2		eliin 3893	. . . 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 3151	. . . . 5 |- ( B C_ C -> ( y e. B -> y e. C ) )
5	4	reximi 2574	. . . 4 |- ( E. x e. A B C_ C -> E. x e. A ( y e. B -> y e. C ) )
6		r19.36av 2628	. . . 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 3163	1 |- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2148   A.wral 2455   E.wrex 2456   _Vcvv 2739   C_ wss 3131   |^|_ciin 3889

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-iin 3891

This theorem is referenced by:  riinm  3961  reliin  4750  cnviinm  5172  iinerm  6609