Theorem sslm 14224

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 3234	. . . . 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 4296	. . 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 14169	. . 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 14169	. . 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 3215	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 2160   A.wral 2468   E.wrex 2469   C_ wss 3144   {copab 4078   ran crn 4645   |` cres 4646   -->wf 5231   ` cfv 5235  (class class class)co 5897   ^pm cpm 6676   CCcc 7840   ZZ>=cuz 9559  TopOnctopon 13987   ~~> tclm 14164

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7933

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-pm 6678  df-top 13975  df-topon 13988  df-lm 14167

This theorem is referenced by: (None)