Theorem iinss 4027

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 2806	. . . 4 |- y e. _V
2		eliin 3980	. . . 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 3222	. . . . 5 |- ( B C_ C -> ( y e. B -> y e. C ) )
5	4	reximi 2630	. . . 4 |- ( E. x e. A B C_ C -> E. x e. A ( y e. B -> y e. C ) )
6		r19.36av 2685	. . . 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 3234	1 |- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2202   A.wral 2511   E.wrex 2512   _Vcvv 2803   C_ wss 3201   |^|_ciin 3976

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-in 3207  df-ss 3214  df-iin 3978

This theorem is referenced by:  riinm  4048  reliin  4855  cnviinm  5285  iinerm  6819