Theorem iinssf 39327

Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypothesis

Ref	Expression
iinssf.1	|- F/_ x C

Assertion

Ref	Expression
iinssf	|- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C )

Proof of Theorem iinssf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 |- y e. _V
2		eliin 4525	. . . 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 3597	. . . . 5 |- ( B C_ C -> ( y e. B -> y e. C ) )
5	4	reximi 3011	. . . 4 |- ( E. x e. A B C_ C -> E. x e. A ( y e. B -> y e. C ) )
6		nfcv 2764	. . . . . 6 |- F/_ x y
7		iinssf.1	. . . . . 6 |- F/_ x C
8	6, 7	nfel 2777	. . . . 5 |- F/ x y e. C
9	8	r19.36vf 39324	. . . 4 |- ( E. x e. A ( y e. B -> y e. C ) -> ( A. x e. A y e. B -> y e. C ) )
10	5, 9	syl 17	. . 3 |- ( E. x e. A B C_ C -> ( A. x e. A y e. B -> y e. C ) )
11	3, 10	syl5bi 232	. 2 |- ( E. x e. A B C_ C -> ( y e. |^|_ x e. A B -> y e. C ) )
12	11	ssrdv 3609	1 |- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   F/_wnfc 2751   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-iin 4523

This theorem is referenced by:  iinssdf  39328