Theorem iinss 3304

Description: Subset implication for an indexed intersection. (The proof was 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

Proof of Theorem iinss

Step	Hyp	Ref	Expression
1		ssel 2615	. . . . 5 |- (B C_ C -> (y e. B -> y e. C))
2	1	reximi 2198	. . . 4 |- (E.x e. A B C_ C -> E.x e. A (y e. B -> y e. C))
3		r19.36av 2232	. . . 4 |- (E.x e. A (y e. B -> y e. C) -> (A.x e. A y e. B -> y e. C))
4	2, 3	syl 12	. . 3 |- (E.x e. A B C_ C -> (A.x e. A y e. B -> y e. C))
5		visset 2295	. . . 4 |- y e. _V
6		eliin 3260	. . . 4 |- (y e. _V -> (y e. |^|_x e. A B <-> A.x e. A y e. B))
7	5, 6	ax-mp 7	. . 3 |- (y e. |^|_x e. A B <-> A.x e. A y e. B)
8	4, 7	syl5ib 223	. 2 |- (E.x e. A B C_ C -> (y e. |^|_x e. A B -> y e. C))
9	8	ssrdv 2622	1 |- (E.x e. A B C_ C -> |^|_x e. A B C_ C)

Syntax hints:   -> wi 3   <-> wb 163   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  |^|_ciin 3256

This theorem is referenced by:  scott0 5847  inpreima2 14468  iintlem2 15011  isfclus 15606  pmapglb2 17253  pmapglb2x 17254

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-in 2603  df-ss 2605  df-iin 3258

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