Theorem submre 16265

Description: The subcollection of a closed set system below a given closed set is itself a closed set system. (Contributed by Stefan O'Rear, 9-Mar-2015.)

Assertion

Ref	Expression
submre	|- ( ( C e. (Moore ` X ) /\ A e. C ) -> ( C i^i ~P A ) e. (Moore ` A ) )

Proof of Theorem submre

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 3834	. . 3 |- ( C i^i ~P A ) C_ ~P A
2	1	a1i 11	. 2 |- ( ( C e. (Moore ` X ) /\ A e. C ) -> ( C i^i ~P A ) C_ ~P A )
3		simpr 477	. . 3 |- ( ( C e. (Moore ` X ) /\ A e. C ) -> A e. C )
4		pwidg 4173	. . . 4 |- ( A e. C -> A e. ~P A )
5	4	adantl 482	. . 3 |- ( ( C e. (Moore ` X ) /\ A e. C ) -> A e. ~P A )
6	3, 5	elind 3798	. 2 |- ( ( C e. (Moore ` X ) /\ A e. C ) -> A e. ( C i^i ~P A ) )
7		simp1l 1085	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ A e. C ) /\ x C_ ( C i^i ~P A ) /\ x =/= (/) ) -> C e. (Moore ` X ) )
8		inss1 3833	. . . . . 6 |- ( C i^i ~P A ) C_ C
9		sstr 3611	. . . . . 6 |- ( ( x C_ ( C i^i ~P A ) /\ ( C i^i ~P A ) C_ C ) -> x C_ C )
10	8, 9	mpan2 707	. . . . 5 |- ( x C_ ( C i^i ~P A ) -> x C_ C )
11	10	3ad2ant2 1083	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ A e. C ) /\ x C_ ( C i^i ~P A ) /\ x =/= (/) ) -> x C_ C )
12		simp3 1063	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ A e. C ) /\ x C_ ( C i^i ~P A ) /\ x =/= (/) ) -> x =/= (/) )
13		mreintcl 16255	. . . 4 |- ( ( C e. (Moore ` X ) /\ x C_ C /\ x =/= (/) ) -> |^| x e. C )
14	7, 11, 12, 13	syl3anc 1326	. . 3 |- ( ( ( C e. (Moore ` X ) /\ A e. C ) /\ x C_ ( C i^i ~P A ) /\ x =/= (/) ) -> |^| x e. C )
15		sstr 3611	. . . . . . . 8 |- ( ( x C_ ( C i^i ~P A ) /\ ( C i^i ~P A ) C_ ~P A ) -> x C_ ~P A )
16	1, 15	mpan2 707	. . . . . . 7 |- ( x C_ ( C i^i ~P A ) -> x C_ ~P A )
17	16	3ad2ant2 1083	. . . . . 6 |- ( ( ( C e. (Moore ` X ) /\ A e. C ) /\ x C_ ( C i^i ~P A ) /\ x =/= (/) ) -> x C_ ~P A )
18		intssuni2 4502	. . . . . 6 |- ( ( x C_ ~P A /\ x =/= (/) ) -> |^| x C_ U. ~P A )
19	17, 12, 18	syl2anc 693	. . . . 5 |- ( ( ( C e. (Moore ` X ) /\ A e. C ) /\ x C_ ( C i^i ~P A ) /\ x =/= (/) ) -> |^| x C_ U. ~P A )
20		unipw 4918	. . . . 5 |- U. ~P A = A
21	19, 20	syl6sseq 3651	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ A e. C ) /\ x C_ ( C i^i ~P A ) /\ x =/= (/) ) -> |^| x C_ A )
22		elpw2g 4827	. . . . . 6 |- ( A e. C -> ( |^| x e. ~P A <-> |^| x C_ A ) )
23	22	adantl 482	. . . . 5 |- ( ( C e. (Moore ` X ) /\ A e. C ) -> ( |^| x e. ~P A <-> |^| x C_ A ) )
24	23	3ad2ant1 1082	. . . 4 |- ( ( ( C e. (Moore ` X ) /\ A e. C ) /\ x C_ ( C i^i ~P A ) /\ x =/= (/) ) -> ( |^| x e. ~P A <-> |^| x C_ A ) )
25	21, 24	mpbird 247	. . 3 |- ( ( ( C e. (Moore ` X ) /\ A e. C ) /\ x C_ ( C i^i ~P A ) /\ x =/= (/) ) -> |^| x e. ~P A )
26	14, 25	elind 3798	. 2 |- ( ( ( C e. (Moore ` X ) /\ A e. C ) /\ x C_ ( C i^i ~P A ) /\ x =/= (/) ) -> |^| x e. ( C i^i ~P A ) )
27	2, 6, 26	ismred 16262	1 |- ( ( C e. (Moore ` X ) /\ A e. C ) -> ( C i^i ~P A ) e. (Moore ` A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|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:  submrc  16288