Theorem sslm 14426

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 3244	. . . . 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 1330	. . . 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 4310	. . 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 1022	. 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 14371	. . 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 1021	. 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 14371	. . 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 1020	. 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 3225	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( ~~> t ` K ) C_ ( ~~> t ` J ) )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   C_ wss 3154   {copab 4090   ran crn 4661   |` cres 4662   -->wf 5251   ` cfv 5255  (class class class)co 5919   ^pm cpm 6705   CCcc 7872   ZZ>=cuz 9595  TopOnctopon 14189   ~~> tclm 14366

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-pm 6707  df-top 14177  df-topon 14190  df-lm 14369

This theorem is referenced by: (None)