Theorem sslm 14936

Description: A finer topology has fewer convergent sequences (but the sequences that do converge, converge to the same value). (Contributed by Mario Carneiro, 15-Sep-2015.)

Assertion

Ref	Expression
sslm	|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( ~~> t ` K ) C_ ( ~~> t ` J ) )

Proof of Theorem sslm

Dummy variables u f x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idd 21	. . . . 5 |- ( J C_ K -> ( f e. ( X ^pm CC ) -> f e. ( X ^pm CC ) ) )
2		idd 21	. . . . 5 |- ( J C_ K -> ( x e. X -> x e. X ) )
3		ssralv 3288	. . . . 5 |- ( J C_ K -> ( A. u e. K ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) -> A. u e. J ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) )
4	1, 2, 3	3anim123d 1353	. . . 4 |- ( J C_ K -> ( ( f e. ( X ^pm CC ) /\ x e. X /\ A. u e. K ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) -> ( f e. ( X ^pm CC ) /\ x e. X /\ A. u e. J ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) ) )
5	4	ssopab2dv 4367	. . 3 |- ( J C_ K -> { <. f , x >. | ( f e. ( X ^pm CC ) /\ x e. X /\ A. u e. K ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } C_ { <. f , x >. | ( f e. ( X ^pm CC ) /\ x e. X /\ A. u e. J ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )
6	5	3ad2ant3 1044	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> { <. f , x >. | ( f e. ( X ^pm CC ) /\ x e. X /\ A. u e. K ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } C_ { <. f , x >. | ( f e. ( X ^pm CC ) /\ x e. X /\ A. u e. J ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )
7		lmfval 14882	. . 3 |- ( K e. (TopOn ` X ) -> ( ~~> t ` K ) = { <. f , x >. | ( f e. ( X ^pm CC ) /\ x e. X /\ A. u e. K ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )
8	7	3ad2ant2 1043	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( ~~> t ` K ) = { <. f , x >. | ( f e. ( X ^pm CC ) /\ x e. X /\ A. u e. K ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )
9		lmfval 14882	. . 3 |- ( J e. (TopOn ` X ) -> ( ~~> t ` J ) = { <. f , x >. | ( f e. ( X ^pm CC ) /\ x e. X /\ A. u e. J ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )
10	9	3ad2ant1 1042	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( ~~> t ` J ) = { <. f , x >. | ( f e. ( X ^pm CC ) /\ x e. X /\ A. u e. J ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )
11	6, 8, 10	3sstr4d 3269	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( ~~> t ` K ) C_ ( ~~> t ` J ) )

Syntax hints:   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   C_ wss 3197   {copab 4144   ran crn 4720   |` cres 4721   -->wf 5314   ` cfv 5318  (class class class)co 6007   ^pm cpm 6804   CCcc 8008   ZZ>=cuz 9733  TopOnctopon 14699   ~~> tclm 14876

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-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-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8101

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  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-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  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-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pm 6806  df-top 14687  df-topon 14700  df-lm 14879

This theorem is referenced by: (None)