Theorem cnss1 14900

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 2229	. . . . . 6 |- U. L = U. L
3	1, 2	cnf 14878	. . . . 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 14894	. . . . . . 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 3225	. . . . 5 |- ( ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) /\ x e. L ) -> ( `' f " x ) e. K )
9	8	ralrimiva 2603	. . . 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 14876	. . . . . . 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 14692	. . . . . 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 14871	. . . . 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 950	. . 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 3230	1 |- ( ( K e. (TopOn ` X ) /\ J C_ K ) -> ( J Cn L ) C_ ( K Cn L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   U.cuni 3888   `'ccnv 4718   "cima 4722   -->wf 5314   ` cfv 5318  (class class class)co 6001   Topctop 14671  TopOnctopon 14684   Cn ccn 14859

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-map 6797  df-top 14672  df-topon 14685  df-cn 14862

This theorem is referenced by: (None)