Theorem cnpflf 21805

Description: Continuity of a function at a point in terms of filter limits. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)

Assertion

Ref	Expression
cnpflf	|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) -> ( F e. ( ( J CnP K ) ` A ) <-> ( F : X --> Y /\ A. f e. ( Fil ` X ) ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) ) ) )

Distinct variable groups:   A, f   f, X   f, Y   f, F   f, J   f, K

Proof of Theorem cnpflf

Step	Hyp	Ref	Expression
1		cnpf2 21054	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( ( J CnP K ) ` A ) ) -> F : X --> Y )
2	1	3expa 1265	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) /\ F e. ( ( J CnP K ) ` A ) ) -> F : X --> Y )
3	2	3adantl3 1219	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ F e. ( ( J CnP K ) ` A ) ) -> F : X --> Y )
4		cnpflfi 21803	. . . . . . 7 |- ( ( A e. ( J fLim f ) /\ F e. ( ( J CnP K ) ` A ) ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) )
5	4	expcom 451	. . . . . 6 |- ( F e. ( ( J CnP K ) ` A ) -> ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) )
6	5	ralrimivw 2967	. . . . 5 |- ( F e. ( ( J CnP K ) ` A ) -> A. f e. ( Fil ` X ) ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) )
7	6	adantl 482	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ F e. ( ( J CnP K ) ` A ) ) -> A. f e. ( Fil ` X ) ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) )
8	3, 7	jca 554	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ F e. ( ( J CnP K ) ` A ) ) -> ( F : X --> Y /\ A. f e. ( Fil ` X ) ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) ) )
9	8	ex 450	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) -> ( F e. ( ( J CnP K ) ` A ) -> ( F : X --> Y /\ A. f e. ( Fil ` X ) ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) ) ) )
10		simpl1 1064	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ F : X --> Y ) -> J e. (TopOn ` X ) )
11		simpl3 1066	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ F : X --> Y ) -> A e. X )
12		neiflim 21778	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A e. X ) -> A e. ( J fLim ( ( nei ` J ) ` { A } ) ) )
13	10, 11, 12	syl2anc 693	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ F : X --> Y ) -> A e. ( J fLim ( ( nei ` J ) ` { A } ) ) )
14	11	snssd 4340	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ F : X --> Y ) -> { A } C_ X )
15		snnzg 4308	. . . . . . . 8 |- ( A e. X -> { A } =/= (/) )
16	11, 15	syl 17	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ F : X --> Y ) -> { A } =/= (/) )
17		neifil 21684	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ { A } C_ X /\ { A } =/= (/) ) -> ( ( nei ` J ) ` { A } ) e. ( Fil ` X ) )
18	10, 14, 16, 17	syl3anc 1326	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ F : X --> Y ) -> ( ( nei ` J ) ` { A } ) e. ( Fil ` X ) )
19		oveq2 6658	. . . . . . . . 9 |- ( f = ( ( nei ` J ) ` { A } ) -> ( J fLim f ) = ( J fLim ( ( nei ` J ) ` { A } ) ) )
20	19	eleq2d 2687	. . . . . . . 8 |- ( f = ( ( nei ` J ) ` { A } ) -> ( A e. ( J fLim f ) <-> A e. ( J fLim ( ( nei ` J ) ` { A } ) ) ) )
21		oveq2 6658	. . . . . . . . . 10 |- ( f = ( ( nei ` J ) ` { A } ) -> ( K fLimf f ) = ( K fLimf ( ( nei ` J ) ` { A } ) ) )
22	21	fveq1d 6193	. . . . . . . . 9 |- ( f = ( ( nei ` J ) ` { A } ) -> ( ( K fLimf f ) ` F ) = ( ( K fLimf ( ( nei ` J ) ` { A } ) ) ` F ) )
23	22	eleq2d 2687	. . . . . . . 8 |- ( f = ( ( nei ` J ) ` { A } ) -> ( ( F ` A ) e. ( ( K fLimf f ) ` F ) <-> ( F ` A ) e. ( ( K fLimf ( ( nei ` J ) ` { A } ) ) ` F ) ) )
24	20, 23	imbi12d 334	. . . . . . 7 |- ( f = ( ( nei ` J ) ` { A } ) -> ( ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) <-> ( A e. ( J fLim ( ( nei ` J ) ` { A } ) ) -> ( F ` A ) e. ( ( K fLimf ( ( nei ` J ) ` { A } ) ) ` F ) ) ) )
25	24	rspcv 3305	. . . . . 6 |- ( ( ( nei ` J ) ` { A } ) e. ( Fil ` X ) -> ( A. f e. ( Fil ` X ) ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) -> ( A e. ( J fLim ( ( nei ` J ) ` { A } ) ) -> ( F ` A ) e. ( ( K fLimf ( ( nei ` J ) ` { A } ) ) ` F ) ) ) )
26	18, 25	syl 17	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ F : X --> Y ) -> ( A. f e. ( Fil ` X ) ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) -> ( A e. ( J fLim ( ( nei ` J ) ` { A } ) ) -> ( F ` A ) e. ( ( K fLimf ( ( nei ` J ) ` { A } ) ) ` F ) ) ) )
27	13, 26	mpid 44	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) /\ F : X --> Y ) -> ( A. f e. ( Fil ` X ) ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) -> ( F ` A ) e. ( ( K fLimf ( ( nei ` J ) ` { A } ) ) ` F ) ) )
28	27	imdistanda 729	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) -> ( ( F : X --> Y /\ A. f e. ( Fil ` X ) ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) ) -> ( F : X --> Y /\ ( F ` A ) e. ( ( K fLimf ( ( nei ` J ) ` { A } ) ) ` F ) ) ) )
29		eqid 2622	. . . 4 |- ( ( nei ` J ) ` { A } ) = ( ( nei ` J ) ` { A } )
30	29	cnpflf2 21804	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) -> ( F e. ( ( J CnP K ) ` A ) <-> ( F : X --> Y /\ ( F ` A ) e. ( ( K fLimf ( ( nei ` J ) ` { A } ) ) ` F ) ) ) )
31	28, 30	sylibrd 249	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) -> ( ( F : X --> Y /\ A. f e. ( Fil ` X ) ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) ) -> F e. ( ( J CnP K ) ` A ) ) )
32	9, 31	impbid 202	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ A e. X ) -> ( F e. ( ( J CnP K ) ` A ) <-> ( F : X --> Y /\ A. f e. ( Fil ` X ) ( A e. ( J fLim f ) -> ( F ` A ) e. ( ( K fLimf f ) ` F ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   {csn 4177   -->wf 5884   ` cfv 5888  (class class class)co 6650  TopOnctopon 20715   neicnei 20901   CnP ccnp 21029   Filcfil 21649   fLim cflim 21738   fLimf cflf 21739

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-1st 7168  df-2nd 7169  df-map 7859  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-ntr 20824  df-nei 20902  df-cnp 21032  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744

This theorem is referenced by:  cnflf  21806  cnpfcf  21845