Theorem clsint2 32324

Description: The closure of an intersection is a subset of the intersection of the closures. (Contributed by Jeff Hankins, 31-Aug-2009.)

Hypothesis

Ref	Expression
clsint2.1	|- X = U. J

Assertion

Ref	Expression
clsint2	|- ( ( J e. Top /\ C C_ ~P X ) -> ( ( cls ` J ) ` |^| C ) C_ |^|_ c e. C ( ( cls ` J ) ` c ) )

Distinct variable groups:   C, c   J, c   X, c

Proof of Theorem clsint2

Step	Hyp	Ref	Expression
1		sspwuni 4611	. . . 4 |- ( C C_ ~P X <-> U. C C_ X )
2		elssuni 4467	. . . . . . . 8 |- ( c e. C -> c C_ U. C )
3		sstr2 3610	. . . . . . . 8 |- ( c C_ U. C -> ( U. C C_ X -> c C_ X ) )
4	2, 3	syl 17	. . . . . . 7 |- ( c e. C -> ( U. C C_ X -> c C_ X ) )
5	4	adantl 482	. . . . . 6 |- ( ( J e. Top /\ c e. C ) -> ( U. C C_ X -> c C_ X ) )
6		intss1 4492	. . . . . . . . 9 |- ( c e. C -> |^| C C_ c )
7		clsint2.1	. . . . . . . . . 10 |- X = U. J
8	7	clsss 20858	. . . . . . . . 9 |- ( ( J e. Top /\ c C_ X /\ |^| C C_ c ) -> ( ( cls ` J ) ` |^| C ) C_ ( ( cls ` J ) ` c ) )
9	6, 8	syl3an3 1361	. . . . . . . 8 |- ( ( J e. Top /\ c C_ X /\ c e. C ) -> ( ( cls ` J ) ` |^| C ) C_ ( ( cls ` J ) ` c ) )
10	9	3com23 1271	. . . . . . 7 |- ( ( J e. Top /\ c e. C /\ c C_ X ) -> ( ( cls ` J ) ` |^| C ) C_ ( ( cls ` J ) ` c ) )
11	10	3expia 1267	. . . . . 6 |- ( ( J e. Top /\ c e. C ) -> ( c C_ X -> ( ( cls ` J ) ` |^| C ) C_ ( ( cls ` J ) ` c ) ) )
12	5, 11	syld 47	. . . . 5 |- ( ( J e. Top /\ c e. C ) -> ( U. C C_ X -> ( ( cls ` J ) ` |^| C ) C_ ( ( cls ` J ) ` c ) ) )
13	12	impancom 456	. . . 4 |- ( ( J e. Top /\ U. C C_ X ) -> ( c e. C -> ( ( cls ` J ) ` |^| C ) C_ ( ( cls ` J ) ` c ) ) )
14	1, 13	sylan2b 492	. . 3 |- ( ( J e. Top /\ C C_ ~P X ) -> ( c e. C -> ( ( cls ` J ) ` |^| C ) C_ ( ( cls ` J ) ` c ) ) )
15	14	ralrimiv 2965	. 2 |- ( ( J e. Top /\ C C_ ~P X ) -> A. c e. C ( ( cls ` J ) ` |^| C ) C_ ( ( cls ` J ) ` c ) )
16		ssiin 4570	. 2 |- ( ( ( cls ` J ) ` |^| C ) C_ |^|_ c e. C ( ( cls ` J ) ` c ) <-> A. c e. C ( ( cls ` J ) ` |^| C ) C_ ( ( cls ` J ) ` c ) )
17	15, 16	sylibr 224	1 |- ( ( J e. Top /\ C C_ ~P X ) -> ( ( cls ` J ) ` |^| C ) C_ |^|_ c e. C ( ( cls ` J ) ` c ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   |^|_ciin 4521   ` cfv 5888   Topctop 20698   clsccl 20822

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-cld 20823  df-cls 20825

This theorem is referenced by: (None)