Theorem cnss2 15218

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 2234	. . . . . 6 |- U. J = U. J
2		cnss2.1	. . . . . 6 |- Y = U. K
3	1, 2	cnf 15195	. . . . 5 |- ( f e. ( J Cn K ) -> f : U. J --> Y )
4	3	adantl 277	. . . 4 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> f : U. J --> Y )
5		simplr 529	. . . . 5 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> L C_ K )
6		cnima 15211	. . . . . . 7 |- ( ( f e. ( J Cn K ) /\ x e. K ) -> ( `' f " x ) e. J )
7	6	ralrimiva 2617	. . . . . 6 |- ( f e. ( J Cn K ) -> A. x e. K ( `' f " x ) e. J )
8	7	adantl 277	. . . . 5 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> A. x e. K ( `' f " x ) e. J )
9		ssralv 3306	. . . . 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 15192	. . . . . . 7 |- ( f e. ( J Cn K ) -> J e. Top )
12	11	adantl 277	. . . . . 6 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> J e. Top )
13	1	toptopon 15009	. . . . . 6 |- ( J e. Top <-> J e. (TopOn ` U. J ) )
14	12, 13	sylib 122	. . . . 5 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> J e. (TopOn ` U. J ) )
15		simpll 527	. . . . 5 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> L e. (TopOn ` Y ) )
16		iscn 15188	. . . . 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 411	. . . 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 953	. . 3 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> f e. ( J Cn L ) )
19	18	ex 115	. 2 |- ( ( L e. (TopOn ` Y ) /\ L C_ K ) -> ( f e. ( J Cn K ) -> f e. ( J Cn L ) ) )
20	19	ssrdv 3248	1 |- ( ( L e. (TopOn ` Y ) /\ L C_ K ) -> ( J Cn K ) C_ ( J Cn L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   C_ wss 3214   U.cuni 3919   `'ccnv 4753   "cima 4757   -->wf 5353   ` cfv 5357  (class class class)co 6058   Topctop 14988  TopOnctopon 15001   Cn ccn 15176

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-top 14989  df-topon 15002  df-cn 15179

This theorem is referenced by: (None)