Theorem clsss 14792

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 3231	. . . . . 6 |- ( T C_ S -> ( S C_ x -> T C_ x ) )
2	1	adantr 276	. . . . 5 |- ( ( T C_ S /\ x e. ( Clsd ` J ) ) -> ( S C_ x -> T C_ x ) )
3	2	ss2rabdv 3305	. . . 4 |- ( T C_ S -> { x e. ( Clsd ` J ) | S C_ x } C_ { x e. ( Clsd ` J ) | T C_ x } )
4		intss 3944	. . . 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 14	. . 3 |- ( T C_ S -> |^| { x e. ( Clsd ` J ) | T C_ x } C_ |^| { x e. ( Clsd ` J ) | S C_ x } )
6	5	3ad2ant3 1044	. 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 1021	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> J e. Top )
8		sstr2 3231	. . . . 5 |- ( T C_ S -> ( S C_ X -> T C_ X ) )
9	8	impcom 125	. . . 4 |- ( ( S C_ X /\ T C_ S ) -> T C_ X )
10	9	3adant1 1039	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> T C_ X )
11		clscld.1	. . . 4 |- X = U. J
12	11	clsval 14785	. . 3 |- ( ( J e. Top /\ T C_ X ) -> ( ( cls ` J ) ` T ) = |^| { x e. ( Clsd ` J ) | T C_ x } )
13	7, 10, 12	syl2anc 411	. 2 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( cls ` J ) ` T ) = |^| { x e. ( Clsd ` J ) | T C_ x } )
14	11	clsval 14785	. . 3 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )
15	14	3adant3 1041	. 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 3269	1 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( cls ` J ) ` T ) C_ ( ( cls ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   C_ wss 3197   U.cuni 3888   |^|cint 3923   ` cfv 5318   Topctop 14671   Clsdccld 14766   clsccl 14768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-top 14672  df-cld 14769  df-cls 14771

This theorem is referenced by:  clsss2  14803