Theorem iinss 3699

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 2554	. . . 4 |- y e. _V
2		eliin 3653	. . . 4 |- (y e. _V -> (y e. |^|_x e. A B <-> A.x e. A y e. B))
3	1, 2	ax-mp 7	. . 3 |- (y e. |^|_x e. A B <-> A.x e. A y e. B)
4		ssel 2933	. . . . 5 |- (B C_ C -> (y e. B -> y e. C))
5	4	reximi 2410	. . . 4 |- (E.x e. A B C_ C -> E.x e. A (y e. B -> y e. C))
6		r19.36av 2455	. . . 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	syl5bi 141	. 2 |- (E.x e. A B C_ C -> (y e. |^|_x e. A B -> y e. C))
9	8	ssrdv 2945	1 |- (E.x e. A B C_ C -> |^|_x e. A B C_ C)

Syntax hints:   -> wi 4   <-> wb 98   e. wcel 1390  A.wral 2300  E.wrex 2301   _Vcvv 2551   C_ wss 2911   |^|_ciin 3649

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iin 3651

This theorem is referenced by:  riinm  3720  reliin  4402  cnviinm  4802  iinerm  6114