Theorem cnss2 12867

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

Hypothesis

Ref	Expression
cnss2.1	|- Y = U. K

Assertion

Ref	Expression
cnss2	|- ( ( L e. (TopOn ` Y ) /\ L C_ K ) -> ( J Cn K ) C_ ( J Cn L ) )

Proof of Theorem cnss2

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

Step	Hyp	Ref	Expression
1		eqid 2165	. . . . . 6 |- U. J = U. J
2		cnss2.1	. . . . . 6 |- Y = U. K
3	1, 2	cnf 12844	. . . . 5 |- ( f e. ( J Cn K ) -> f : U. J --> Y )
4	3	adantl 275	. . . 4 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> f : U. J --> Y )
5		simplr 520	. . . . 5 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> L C_ K )
6		cnima 12860	. . . . . . 7 |- ( ( f e. ( J Cn K ) /\ x e. K ) -> ( `' f " x ) e. J )
7	6	ralrimiva 2539	. . . . . 6 |- ( f e. ( J Cn K ) -> A. x e. K ( `' f " x ) e. J )
8	7	adantl 275	. . . . 5 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> A. x e. K ( `' f " x ) e. J )
9		ssralv 3206	. . . . 5 |- ( L C_ K -> ( A. x e. K ( `' f " x ) e. J -> A. x e. L ( `' f " x ) e. J ) )
10	5, 8, 9	sylc 62	. . . 4 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> A. x e. L ( `' f " x ) e. J )
11		cntop1 12841	. . . . . . 7 |- ( f e. ( J Cn K ) -> J e. Top )
12	11	adantl 275	. . . . . 6 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> J e. Top )
13	1	toptopon 12656	. . . . . 6 |- ( J e. Top <-> J e. (TopOn ` U. J ) )
14	12, 13	sylib 121	. . . . 5 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> J e. (TopOn ` U. J ) )
15		simpll 519	. . . . 5 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> L e. (TopOn ` Y ) )
16		iscn 12837	. . . . 5 |- ( ( J e. (TopOn ` U. J ) /\ L e. (TopOn ` Y ) ) -> ( f e. ( J Cn L ) <-> ( f : U. J --> Y /\ A. x e. L ( `' f " x ) e. J ) ) )
17	14, 15, 16	syl2anc 409	. . . 4 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> ( f e. ( J Cn L ) <-> ( f : U. J --> Y /\ A. x e. L ( `' f " x ) e. J ) ) )
18	4, 10, 17	mpbir2and 934	. . 3 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> f e. ( J Cn L ) )
19	18	ex 114	. 2 |- ( ( L e. (TopOn ` Y ) /\ L C_ K ) -> ( f e. ( J Cn K ) -> f e. ( J Cn L ) ) )
20	19	ssrdv 3148	1 |- ( ( L e. (TopOn ` Y ) /\ L C_ K ) -> ( J Cn K ) C_ ( J Cn L ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   C_ wss 3116   U.cuni 3789   `'ccnv 4603   "cima 4607   -->wf 5184   ` cfv 5188  (class class class)co 5842   Topctop 12635  TopOnctopon 12648   Cn ccn 12825

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-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-map 6616  df-top 12636  df-topon 12649  df-cn 12828

This theorem is referenced by: (None)