Theorem cnss2 14547

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 2196	. . . . . 6 |- U. J = U. J
2		cnss2.1	. . . . . 6 |- Y = U. K
3	1, 2	cnf 14524	. . . . 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 14540	. . . . . . 7 |- ( ( f e. ( J Cn K ) /\ x e. K ) -> ( `' f " x ) e. J )
7	6	ralrimiva 2570	. . . . . 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 3248	. . . . 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 14521	. . . . . . 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 14338	. . . . . 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 14517	. . . . 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 3190	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 2167   A.wral 2475   C_ wss 3157   U.cuni 3840   `'ccnv 4663   "cima 4667   -->wf 5255   ` cfv 5259  (class class class)co 5925   Topctop 14317  TopOnctopon 14330   Cn ccn 14505

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-top 14318  df-topon 14331  df-cn 14508

This theorem is referenced by: (None)