Theorem clsss 20858

Description: Subset relationship for closure. (Contributed by NM, 10-Feb-2007.)

Hypothesis

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

Assertion

Ref	Expression
clsss	|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( cls ` J ) ` T ) C_ ( ( cls ` J ) ` S ) )

Proof of Theorem clsss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3610	. . . . . 6 |- ( T C_ S -> ( S C_ x -> T C_ x ) )
2	1	adantr 481	. . . . 5 |- ( ( T C_ S /\ x e. ( Clsd ` J ) ) -> ( S C_ x -> T C_ x ) )
3	2	ss2rabdv 3683	. . . 4 |- ( T C_ S -> { x e. ( Clsd ` J ) | S C_ x } C_ { x e. ( Clsd ` J ) | T C_ x } )
4		intss 4498	. . . 4 |- ( { x e. ( Clsd ` J ) | S C_ x } C_ { x e. ( Clsd ` J ) | T C_ x } -> |^| { x e. ( Clsd ` J ) | T C_ x } C_ |^| { x e. ( Clsd ` J ) | S C_ x } )
5	3, 4	syl 17	. . 3 |- ( T C_ S -> |^| { x e. ( Clsd ` J ) | T C_ x } C_ |^| { x e. ( Clsd ` J ) | S C_ x } )
6	5	3ad2ant3 1084	. 2 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> |^| { x e. ( Clsd ` J ) | T C_ x } C_ |^| { x e. ( Clsd ` J ) | S C_ x } )
7		simp1 1061	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> J e. Top )
8		sstr2 3610	. . . . 5 |- ( T C_ S -> ( S C_ X -> T C_ X ) )
9	8	impcom 446	. . . 4 |- ( ( S C_ X /\ T C_ S ) -> T C_ X )
10	9	3adant1 1079	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> T C_ X )
11		clscld.1	. . . 4 |- X = U. J
12	11	clsval 20841	. . 3 |- ( ( J e. Top /\ T C_ X ) -> ( ( cls ` J ) ` T ) = |^| { x e. ( Clsd ` J ) | T C_ x } )
13	7, 10, 12	syl2anc 693	. 2 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( cls ` J ) ` T ) = |^| { x e. ( Clsd ` J ) | T C_ x } )
14	11	clsval 20841	. . 3 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )
15	14	3adant3 1081	. 2 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( cls ` J ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )
16	6, 13, 15	3sstr4d 3648	1 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( cls ` J ) ` T ) C_ ( ( cls ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   U.cuni 4436   |^|cint 4475   ` cfv 5888   Topctop 20698   Clsdccld 20820   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-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:  ntrss  20859  clsss2  20876  lpsscls  20945  lpss3  20948  cnclsi  21076  cncls  21078  lpcls  21168  cnextcn  21871  clssubg  21912  clsnsg  21913  utopreg  22056  hauseqcn  29941  kur14lem6  31193  clsint2  32324  opnregcld  32325