Theorem mrerintcl 16257

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

Assertion

Ref	Expression
mrerintcl	|- ( ( C e. (Moore ` X ) /\ S C_ C ) -> ( X i^i |^| S ) e. C )

Proof of Theorem mrerintcl

Step	Hyp	Ref	Expression
1		rint0 4517	. . . 4 |- ( S = (/) -> ( X i^i |^| S ) = X )
2	1	adantl 482	. . 3 |- ( ( ( C e. (Moore ` X ) /\ S C_ C ) /\ S = (/) ) -> ( X i^i |^| S ) = X )
3		mre1cl 16254	. . . 4 |- ( C e. (Moore ` X ) -> X e. C )
4	3	ad2antrr 762	. . 3 |- ( ( ( C e. (Moore ` X ) /\ S C_ C ) /\ S = (/) ) -> X e. C )
5	2, 4	eqeltrd 2701	. 2 |- ( ( ( C e. (Moore ` X ) /\ S C_ C ) /\ S = (/) ) -> ( X i^i |^| S ) e. C )
6		simp2 1062	. . . . . 6 |- ( ( C e. (Moore ` X ) /\ S C_ C /\ S =/= (/) ) -> S C_ C )
7		mresspw 16252	. . . . . . 7 |- ( C e. (Moore ` X ) -> C C_ ~P X )
8	7	3ad2ant1 1082	. . . . . 6 |- ( ( C e. (Moore ` X ) /\ S C_ C /\ S =/= (/) ) -> C C_ ~P X )
9	6, 8	sstrd 3613	. . . . 5 |- ( ( C e. (Moore ` X ) /\ S C_ C /\ S =/= (/) ) -> S C_ ~P X )
10		simp3 1063	. . . . 5 |- ( ( C e. (Moore ` X ) /\ S C_ C /\ S =/= (/) ) -> S =/= (/) )
11		rintn0 4619	. . . . 5 |- ( ( S C_ ~P X /\ S =/= (/) ) -> ( X i^i |^| S ) = |^| S )
12	9, 10, 11	syl2anc 693	. . . 4 |- ( ( C e. (Moore ` X ) /\ S C_ C /\ S =/= (/) ) -> ( X i^i |^| S ) = |^| S )
13		mreintcl 16255	. . . 4 |- ( ( C e. (Moore ` X ) /\ S C_ C /\ S =/= (/) ) -> |^| S e. C )
14	12, 13	eqeltrd 2701	. . 3 |- ( ( C e. (Moore ` X ) /\ S C_ C /\ S =/= (/) ) -> ( X i^i |^| S ) e. C )
15	14	3expa 1265	. 2 |- ( ( ( C e. (Moore ` X ) /\ S C_ C ) /\ S =/= (/) ) -> ( X i^i |^| S ) e. C )
16	5, 15	pm2.61dane 2881	1 |- ( ( C e. (Moore ` X ) /\ S C_ C ) -> ( X i^i |^| S ) e. C )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |^|cint 4475   ` 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-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:  mreacs  16319  topmtcl  32358