Theorem sslm 13717

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 3219	. . . . 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 1319	. . . 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 4278	. . 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 1020	. 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 13662	. . 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 1019	. 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 13662	. . 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 1018	. 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 3200	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( ~~> t ` K ) C_ ( ~~> t ` J ) )

Syntax hints:   -> wi 4   /\ w3a 978   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   C_ wss 3129   {copab 4063   ran crn 4627   |` cres 4628   -->wf 5212   ` cfv 5216  (class class class)co 5874   ^pm cpm 6648   CCcc 7808   ZZ>=cuz 9527  TopOnctopon 13480   ~~> tclm 13657

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-pm 6650  df-top 13468  df-topon 13481  df-lm 13660

This theorem is referenced by: (None)