Theorem flimcf 21786

Description: Fineness is properly characterized by the property that every limit point of a filter in the finer topology is a limit point in the coarser topology. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
flimcf	|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) -> ( J C_ K <-> A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) )

Distinct variable groups:   f, J   f, K   f, X

Proof of Theorem flimcf

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplll 798	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ J C_ K ) /\ ( f e. ( Fil ` X ) /\ x e. ( K fLim f ) ) ) -> J e. (TopOn ` X ) )
2		simprl 794	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ J C_ K ) /\ ( f e. ( Fil ` X ) /\ x e. ( K fLim f ) ) ) -> f e. ( Fil ` X ) )
3		simplr 792	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ J C_ K ) /\ ( f e. ( Fil ` X ) /\ x e. ( K fLim f ) ) ) -> J C_ K )
4		flimss1 21777	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ f e. ( Fil ` X ) /\ J C_ K ) -> ( K fLim f ) C_ ( J fLim f ) )
5	1, 2, 3, 4	syl3anc 1326	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ J C_ K ) /\ ( f e. ( Fil ` X ) /\ x e. ( K fLim f ) ) ) -> ( K fLim f ) C_ ( J fLim f ) )
6		simprr 796	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ J C_ K ) /\ ( f e. ( Fil ` X ) /\ x e. ( K fLim f ) ) ) -> x e. ( K fLim f ) )
7	5, 6	sseldd 3604	. . . . 5 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ J C_ K ) /\ ( f e. ( Fil ` X ) /\ x e. ( K fLim f ) ) ) -> x e. ( J fLim f ) )
8	7	expr 643	. . . 4 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ J C_ K ) /\ f e. ( Fil ` X ) ) -> ( x e. ( K fLim f ) -> x e. ( J fLim f ) ) )
9	8	ssrdv 3609	. . 3 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ J C_ K ) /\ f e. ( Fil ` X ) ) -> ( K fLim f ) C_ ( J fLim f ) )
10	9	ralrimiva 2966	. 2 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ J C_ K ) -> A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) )
11		simpllr 799	. . . . . . . . . . . 12 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> K e. (TopOn ` X ) )
12		simplll 798	. . . . . . . . . . . . . . 15 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> J e. (TopOn ` X ) )
13		simprl 794	. . . . . . . . . . . . . . 15 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> x e. J )
14		toponss 20731	. . . . . . . . . . . . . . 15 |- ( ( J e. (TopOn ` X ) /\ x e. J ) -> x C_ X )
15	12, 13, 14	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> x C_ X )
16		simprr 796	. . . . . . . . . . . . . 14 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> y e. x )
17	15, 16	sseldd 3604	. . . . . . . . . . . . 13 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> y e. X )
18	17	snssd 4340	. . . . . . . . . . . 12 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> { y } C_ X )
19		snnzg 4308	. . . . . . . . . . . . 13 |- ( y e. X -> { y } =/= (/) )
20	17, 19	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> { y } =/= (/) )
21		neifil 21684	. . . . . . . . . . . 12 |- ( ( K e. (TopOn ` X ) /\ { y } C_ X /\ { y } =/= (/) ) -> ( ( nei ` K ) ` { y } ) e. ( Fil ` X ) )
22	11, 18, 20, 21	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> ( ( nei ` K ) ` { y } ) e. ( Fil ` X ) )
23		simplr 792	. . . . . . . . . . 11 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) )
24		oveq2 6658	. . . . . . . . . . . . 13 |- ( f = ( ( nei ` K ) ` { y } ) -> ( K fLim f ) = ( K fLim ( ( nei ` K ) ` { y } ) ) )
25		oveq2 6658	. . . . . . . . . . . . 13 |- ( f = ( ( nei ` K ) ` { y } ) -> ( J fLim f ) = ( J fLim ( ( nei ` K ) ` { y } ) ) )
26	24, 25	sseq12d 3634	. . . . . . . . . . . 12 |- ( f = ( ( nei ` K ) ` { y } ) -> ( ( K fLim f ) C_ ( J fLim f ) <-> ( K fLim ( ( nei ` K ) ` { y } ) ) C_ ( J fLim ( ( nei ` K ) ` { y } ) ) ) )
27	26	rspcv 3305	. . . . . . . . . . 11 |- ( ( ( nei ` K ) ` { y } ) e. ( Fil ` X ) -> ( A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) -> ( K fLim ( ( nei ` K ) ` { y } ) ) C_ ( J fLim ( ( nei ` K ) ` { y } ) ) ) )
28	22, 23, 27	sylc 65	. . . . . . . . . 10 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> ( K fLim ( ( nei ` K ) ` { y } ) ) C_ ( J fLim ( ( nei ` K ) ` { y } ) ) )
29		neiflim 21778	. . . . . . . . . . 11 |- ( ( K e. (TopOn ` X ) /\ y e. X ) -> y e. ( K fLim ( ( nei ` K ) ` { y } ) ) )
30	11, 17, 29	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> y e. ( K fLim ( ( nei ` K ) ` { y } ) ) )
31	28, 30	sseldd 3604	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> y e. ( J fLim ( ( nei ` K ) ` { y } ) ) )
32		flimneiss 21770	. . . . . . . . 9 |- ( y e. ( J fLim ( ( nei ` K ) ` { y } ) ) -> ( ( nei ` J ) ` { y } ) C_ ( ( nei ` K ) ` { y } ) )
33	31, 32	syl 17	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> ( ( nei ` J ) ` { y } ) C_ ( ( nei ` K ) ` { y } ) )
34		topontop 20718	. . . . . . . . . 10 |- ( J e. (TopOn ` X ) -> J e. Top )
35	12, 34	syl 17	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> J e. Top )
36		opnneip 20923	. . . . . . . . 9 |- ( ( J e. Top /\ x e. J /\ y e. x ) -> x e. ( ( nei ` J ) ` { y } ) )
37	35, 13, 16, 36	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> x e. ( ( nei ` J ) ` { y } ) )
38	33, 37	sseldd 3604	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ ( x e. J /\ y e. x ) ) -> x e. ( ( nei ` K ) ` { y } ) )
39	38	anassrs 680	. . . . . 6 |- ( ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ x e. J ) /\ y e. x ) -> x e. ( ( nei ` K ) ` { y } ) )
40	39	ralrimiva 2966	. . . . 5 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ x e. J ) -> A. y e. x x e. ( ( nei ` K ) ` { y } ) )
41		simpllr 799	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ x e. J ) -> K e. (TopOn ` X ) )
42		topontop 20718	. . . . . 6 |- ( K e. (TopOn ` X ) -> K e. Top )
43		opnnei 20924	. . . . . 6 |- ( K e. Top -> ( x e. K <-> A. y e. x x e. ( ( nei ` K ) ` { y } ) ) )
44	41, 42, 43	3syl 18	. . . . 5 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ x e. J ) -> ( x e. K <-> A. y e. x x e. ( ( nei ` K ) ` { y } ) ) )
45	40, 44	mpbird 247	. . . 4 |- ( ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) /\ x e. J ) -> x e. K )
46	45	ex 450	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) -> ( x e. J -> x e. K ) )
47	46	ssrdv 3609	. 2 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) -> J C_ K )
48	10, 47	impbida 877	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) -> ( J C_ K <-> A. f e. ( Fil ` X ) ( K fLim f ) C_ ( J fLim f ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   {csn 4177   ` cfv 5888  (class class class)co 6650   Topctop 20698  TopOnctopon 20715   neicnei 20901   Filcfil 21649   fLim cflim 21738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-top 20699  df-topon 20716  df-ntr 20824  df-nei 20902  df-fil 21650  df-flim 21743

This theorem is referenced by: (None)