Theorem mreriincl 16258

Description: The relative intersection of a family of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
mreriincl	|- ( ( C e. (Moore ` X ) /\ A. y e. I S e. C ) -> ( X i^i |^|_ y e. I S ) e. C )

Distinct variable groups:   y, I   y, X   y, C

Allowed substitution hint:   S( y)

Proof of Theorem mreriincl

Step	Hyp	Ref	Expression
1		riin0 4594	. . . 4 |- ( I = (/) -> ( X i^i |^|_ y e. I S ) = X )
2	1	adantl 482	. . 3 |- ( ( ( C e. (Moore ` X ) /\ A. y e. I S e. C ) /\ I = (/) ) -> ( X i^i |^|_ y e. I S ) = X )
3		mre1cl 16254	. . . 4 |- ( C e. (Moore ` X ) -> X e. C )
4	3	ad2antrr 762	. . 3 |- ( ( ( C e. (Moore ` X ) /\ A. y e. I S e. C ) /\ I = (/) ) -> X e. C )
5	2, 4	eqeltrd 2701	. 2 |- ( ( ( C e. (Moore ` X ) /\ A. y e. I S e. C ) /\ I = (/) ) -> ( X i^i |^|_ y e. I S ) e. C )
6		mress 16253	. . . . . . 7 |- ( ( C e. (Moore ` X ) /\ S e. C ) -> S C_ X )
7	6	ex 450	. . . . . 6 |- ( C e. (Moore ` X ) -> ( S e. C -> S C_ X ) )
8	7	ralimdv 2963	. . . . 5 |- ( C e. (Moore ` X ) -> ( A. y e. I S e. C -> A. y e. I S C_ X ) )
9	8	imp 445	. . . 4 |- ( ( C e. (Moore ` X ) /\ A. y e. I S e. C ) -> A. y e. I S C_ X )
10		riinn0 4595	. . . 4 |- ( ( A. y e. I S C_ X /\ I =/= (/) ) -> ( X i^i |^|_ y e. I S ) = |^|_ y e. I S )
11	9, 10	sylan 488	. . 3 |- ( ( ( C e. (Moore ` X ) /\ A. y e. I S e. C ) /\ I =/= (/) ) -> ( X i^i |^|_ y e. I S ) = |^|_ y e. I S )
12		simpll 790	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ A. y e. I S e. C ) /\ I =/= (/) ) -> C e. (Moore ` X ) )
13		simpr 477	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ A. y e. I S e. C ) /\ I =/= (/) ) -> I =/= (/) )
14		simplr 792	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ A. y e. I S e. C ) /\ I =/= (/) ) -> A. y e. I S e. C )
15		mreiincl 16256	. . . 4 |- ( ( C e. (Moore ` X ) /\ I =/= (/) /\ A. y e. I S e. C ) -> |^|_ y e. I S e. C )
16	12, 13, 14, 15	syl3anc 1326	. . 3 |- ( ( ( C e. (Moore ` X ) /\ A. y e. I S e. C ) /\ I =/= (/) ) -> |^|_ y e. I S e. C )
17	11, 16	eqeltrd 2701	. 2 |- ( ( ( C e. (Moore ` X ) /\ A. y e. I S e. C ) /\ I =/= (/) ) -> ( X i^i |^|_ y e. I S ) e. C )
18	5, 17	pm2.61dane 2881	1 |- ( ( C e. (Moore ` X ) /\ A. y e. I S e. C ) -> ( X i^i |^|_ y e. I S ) e. C )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   i^i cin 3573   C_ wss 3574   (/)c0 3915   |^|_ciin 4521   ` cfv 5888  Moorecmre 16242

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-mre 16246

This theorem is referenced by:  acsfn1  16322  acsfn1c  16323  acsfn2  16324  acsfn1p  37769