Theorem iinss 4018

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 2863	. . . 4 |- y e. _V
2		eliin 3975	. . . 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 3268	. . . . 5 |- (B C_ C -> (y e. B -> y e. C))
5	4	reximi 2722	. . . 4 |- (E.x e. A B C_ C -> E.x e. A (y e. B -> y e. C))
6		r19.36av 2760	. . . 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 15	. . 3 |- (E.x e. A B C_ C -> (A.x e. A y e. B -> y e. C))
8	3, 7	syl5bi 208	. 2 |- (E.x e. A B C_ C -> (y e. |^|_x e. A B -> y e. C))
9	8	ssrdv 3279	1 |- (E.x e. A B C_ C -> |^|_x e. A B C_ C)

Syntax hints:   -> wi 4   <-> wb 176   e. wcel 1710  A.wral 2615  E.wrex 2616  _Vcvv 2860   C_ wss 3258  |^|_ciin 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iin 3973

This theorem is referenced by:  riinn0  4041