Theorem cnss1 13020

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 2170	. . . . . 6 |- U. L = U. L
3	1, 2	cnf 12998	. . . . 5 |- ( f e. ( J Cn L ) -> f : X --> U. L )
4	3	adantl 275	. . . 4 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) -> f : X --> U. L )
5		simpllr 529	. . . . . 6 |- ( ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) /\ x e. L ) -> J C_ K )
6		cnima 13014	. . . . . . 7 |- ( ( f e. ( J Cn L ) /\ x e. L ) -> ( `' f " x ) e. J )
7	6	adantll 473	. . . . . 6 |- ( ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) /\ x e. L ) -> ( `' f " x ) e. J )
8	5, 7	sseldd 3148	. . . . 5 |- ( ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) /\ x e. L ) -> ( `' f " x ) e. K )
9	8	ralrimiva 2543	. . . 4 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) -> A. x e. L ( `' f " x ) e. K )
10		simpll 524	. . . . 5 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) -> K e. (TopOn ` X ) )
11		cntop2 12996	. . . . . . 7 |- ( f e. ( J Cn L ) -> L e. Top )
12	11	adantl 275	. . . . . 6 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) -> L e. Top )
13	2	toptopon 12810	. . . . . 6 |- ( L e. Top <-> L e. (TopOn ` U. L ) )
14	12, 13	sylib 121	. . . . 5 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) -> L e. (TopOn ` U. L ) )
15		iscn 12991	. . . . 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 409	. . . 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 939	. . 3 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ f e. ( J Cn L ) ) -> f e. ( K Cn L ) )
18	17	ex 114	. 2 |- ( ( K e. (TopOn ` X ) /\ J C_ K ) -> ( f e. ( J Cn L ) -> f e. ( K Cn L ) ) )
19	18	ssrdv 3153	1 |- ( ( K e. (TopOn ` X ) /\ J C_ K ) -> ( J Cn L ) C_ ( K Cn L ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   C_ wss 3121   U.cuni 3796   `'ccnv 4610   "cima 4614   -->wf 5194   ` cfv 5198  (class class class)co 5853   Topctop 12789  TopOnctopon 12802   Cn ccn 12979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-top 12790  df-topon 12803  df-cn 12982

This theorem is referenced by: (None)