Theorem cnss1 14405

Description: If the topology K is finer than J, then there are more continuous functions from K than from J. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)

Hypothesis

Ref	Expression
cnss1.1	|- X = U. J

Assertion

Ref	Expression
cnss1	|- ( ( K e. (TopOn ` X ) /\ J C_ K ) -> ( J Cn L ) C_ ( K Cn L ) )

Proof of Theorem cnss1

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

Step	Hyp	Ref	Expression
1		cnss1.1	. . . . . 6 |- X = U. J
2		eqid 2193	. . . . . 6 |- U. L = U. L
3	1, 2	cnf 14383	. . . . 5 |- ( f e. ( J Cn L ) -> f : X --> U. L )
4	3	adantl 277	. . . 4 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) -> f : X --> U. L )
5		simpllr 534	. . . . . 6 |- ( ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) /\ x e. L ) -> J C_ K )
6		cnima 14399	. . . . . . 7 |- ( ( f e. ( J Cn L ) /\ x e. L ) -> ( `' f " x ) e. J )
7	6	adantll 476	. . . . . 6 |- ( ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) /\ x e. L ) -> ( `' f " x ) e. J )
8	5, 7	sseldd 3181	. . . . 5 |- ( ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) /\ x e. L ) -> ( `' f " x ) e. K )
9	8	ralrimiva 2567	. . . 4 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) -> A. x e. L ( `' f " x ) e. K )
10		simpll 527	. . . . 5 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) -> K e. (TopOn ` X ) )
11		cntop2 14381	. . . . . . 7 |- ( f e. ( J Cn L ) -> L e. Top )
12	11	adantl 277	. . . . . 6 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) -> L e. Top )
13	2	toptopon 14197	. . . . . 6 |- ( L e. Top <-> L e. (TopOn ` U. L ) )
14	12, 13	sylib 122	. . . . 5 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) -> L e. (TopOn ` U. L ) )
15		iscn 14376	. . . . 5 |- ( ( K e. (TopOn ` X ) /\ L e. (TopOn ` U. L ) ) -> ( f e. ( K Cn L ) <-> ( f : X --> U. L /\ A. x e. L ( `' f " x ) e. K ) ) )
16	10, 14, 15	syl2anc 411	. . . 4 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) -> ( f e. ( K Cn L ) <-> ( f : X --> U. L /\ A. x e. L ( `' f " x ) e. K ) ) )
17	4, 9, 16	mpbir2and 946	. . 3 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) -> f e. ( K Cn L ) )
18	17	ex 115	. 2 |- ( ( K e. (TopOn ` X ) /\ J C_ K ) -> ( f e. ( J Cn L ) -> f e. ( K Cn L ) ) )
19	18	ssrdv 3186	1 |- ( ( K e. (TopOn ` X ) /\ J C_ K ) -> ( J Cn L ) C_ ( K Cn L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   C_ wss 3154   U.cuni 3836   `'ccnv 4659   "cima 4663   -->wf 5251   ` cfv 5255  (class class class)co 5919   Topctop 14176  TopOnctopon 14189   Cn ccn 14364

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-in1 615  ax-in2 616  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-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  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-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  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-map 6706  df-top 14177  df-topon 14190  df-cn 14367

This theorem is referenced by: (None)