Theorem cnss2 14204

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 2189	. . . . . 6 |- U. J = U. J
2		cnss2.1	. . . . . 6 |- Y = U. K
3	1, 2	cnf 14181	. . . . 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 528	. . . . 5 |- ( ( ( L e. (TopOn ` Y ) /\ L C_ K ) /\ f e. ( J Cn K ) ) -> L C_ K )
6		cnima 14197	. . . . . . 7 |- ( ( f e. ( J Cn K ) /\ x e. K ) -> ( `' f " x ) e. J )
7	6	ralrimiva 2563	. . . . . 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 3234	. . . . 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 14178	. . . . . . 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 13995	. . . . . 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 14174	. . . . 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 946	. . 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 3176	1 |- ( ( L e. (TopOn ` Y ) /\ L C_ K ) -> ( J Cn K ) C_ ( J Cn L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   A.wral 2468   C_ wss 3144   U.cuni 3824   `'ccnv 4643   "cima 4647   -->wf 5231   ` cfv 5235  (class class class)co 5897   Topctop 13974  TopOnctopon 13987   Cn ccn 14162

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  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-map 6677  df-top 13975  df-topon 13988  df-cn 14165

This theorem is referenced by: (None)