Theorem sslm 12788

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 3201	. . . . 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 1308	. . . 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 4250	. . 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 1009	. 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 12733	. . 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 1008	. 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 12733	. . 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 1007	. 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 3182	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( ~~> t ` K ) C_ ( ~~> t ` J ) )

Syntax hints:   -> wi 4   /\ w3a 967   = wceq 1342   e. wcel 2135   A.wral 2442   E.wrex 2443   C_ wss 3111   {copab 4036   ran crn 4599   |` cres 4600   -->wf 5178   ` cfv 5182  (class class class)co 5836   ^pm cpm 6606   CCcc 7742   ZZ>=cuz 9457  TopOnctopon 12549   ~~> tclm 12728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-cnex 7835

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-pm 6608  df-top 12537  df-topon 12550  df-lm 12731

This theorem is referenced by: (None)