Theorem cnss1 13297

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 2175	. . . . . 6 |- U. L = U. L
3	1, 2	cnf 13275	. . . . 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 534	. . . . . 6 |- ( ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) /\ x e. L ) -> J C_ K )
6		cnima 13291	. . . . . . 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 3154	. . . . 5 |- ( ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) /\ x e. L ) -> ( `' f " x ) e. K )
9	8	ralrimiva 2548	. . . 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 13273	. . . . . . 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 13087	. . . . . 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 13268	. . . . 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 944	. . 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 3159	1 |- ( ( K e. (TopOn ` X ) /\ J C_ K ) -> ( J Cn L ) C_ ( K Cn L ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2146   A.wral 2453   C_ wss 3127   U.cuni 3805   `'ccnv 4619   "cima 4623   -->wf 5204   ` cfv 5208  (class class class)co 5865   Topctop 13066  TopOnctopon 13079   Cn ccn 13256

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-map 6640  df-top 13067  df-topon 13080  df-cn 13259

This theorem is referenced by: (None)