Theorem cnss1 14952

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 2231	. . . . . 6 |- U. L = U. L
3	1, 2	cnf 14930	. . . . 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 536	. . . . . 6 |- ( ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) /\ x e. L ) -> J C_ K )
6		cnima 14946	. . . . . . 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 3228	. . . . 5 |- ( ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) /\ x e. L ) -> ( `' f " x ) e. K )
9	8	ralrimiva 2605	. . . 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 14928	. . . . . . 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 14744	. . . . . 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 14923	. . . . 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 952	. . 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 3233	1 |- ( ( K e. (TopOn ` X ) /\ J C_ K ) -> ( J Cn L ) C_ ( K Cn L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   U.cuni 3893   `'ccnv 4724   "cima 4728   -->wf 5322   ` cfv 5326  (class class class)co 6018   Topctop 14723  TopOnctopon 14736   Cn ccn 14911

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-map 6819  df-top 14724  df-topon 14737  df-cn 14914

This theorem is referenced by: (None)