Theorem ssiinf 4569

Description: Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)

Hypothesis

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

Assertion

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

Proof of Theorem ssiinf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 |- y e. _V
2		eliin 4525	. . . . 5 |- ( y e. _V -> ( y e. |^|_ x e. A B <-> A. x e. A y e. B ) )
3	1, 2	ax-mp 5	. . . 4 |- ( y e. |^|_ x e. A B <-> A. x e. A y e. B )
4	3	ralbii 2980	. . 3 |- ( A. y e. C y e. |^|_ x e. A B <-> A. y e. C A. x e. A y e. B )
5		ssiinf.1	. . . 4 |- F/_ x C
6		nfcv 2764	. . . 4 |- F/_ y A
7	5, 6	ralcomf 3096	. . 3 |- ( A. y e. C A. x e. A y e. B <-> A. x e. A A. y e. C y e. B )
8	4, 7	bitri 264	. 2 |- ( A. y e. C y e. |^|_ x e. A B <-> A. x e. A A. y e. C y e. B )
9		dfss3 3592	. 2 |- ( C C_ |^|_ x e. A B <-> A. y e. C y e. |^|_ x e. A B )
10		dfss3 3592	. . 3 |- ( C C_ B <-> A. y e. C y e. B )
11	10	ralbii 2980	. 2 |- ( A. x e. A C C_ B <-> A. x e. A A. y e. C y e. B )
12	8, 9, 11	3bitr4i 292	1 |- ( C C_ |^|_ x e. A B <-> A. x e. A C C_ B )

Syntax hints:   <-> wb 196   e. wcel 1990   F/_wnfc 2751   A.wral 2912   _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-v 3202  df-in 3581  df-ss 3588  df-iin 4523

This theorem is referenced by:  ssiin  4570  dmiin  5369