Theorem flfcntr 21847

Description: A continuous function's value is always in the trace of its filter limit. (Contributed by Thierry Arnoux, 30-Aug-2020.)

Hypotheses

Ref	Expression
flfcntr.c	|- C = U. J
flfcntr.b	|- B = U. K
flfcntr.j	|- ( ph -> J e. Top )
flfcntr.a	|- ( ph -> A C_ C )
flfcntr.1	|- ( ph -> F e. ( ( J ↾t A ) Cn K ) )
flfcntr.y	|- ( ph -> X e. A )

Assertion

Ref	Expression
flfcntr	|- ( ph -> ( F ` X ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) )

Proof of Theorem flfcntr

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

Step	Hyp	Ref	Expression
1		flfcntr.1	. . . . 5 |- ( ph -> F e. ( ( J ↾t A ) Cn K ) )
2		flfcntr.j	. . . . . . . 8 |- ( ph -> J e. Top )
3		flfcntr.c	. . . . . . . . 9 |- C = U. J
4	3	toptopon 20722	. . . . . . . 8 |- ( J e. Top <-> J e. (TopOn ` C ) )
5	2, 4	sylib 208	. . . . . . 7 |- ( ph -> J e. (TopOn ` C ) )
6		flfcntr.a	. . . . . . 7 |- ( ph -> A C_ C )
7		resttopon 20965	. . . . . . 7 |- ( ( J e. (TopOn ` C ) /\ A C_ C ) -> ( J ↾t A ) e. (TopOn ` A ) )
8	5, 6, 7	syl2anc 693	. . . . . 6 |- ( ph -> ( J ↾t A ) e. (TopOn ` A ) )
9		cntop2 21045	. . . . . . . 8 |- ( F e. ( ( J ↾t A ) Cn K ) -> K e. Top )
10	1, 9	syl 17	. . . . . . 7 |- ( ph -> K e. Top )
11		flfcntr.b	. . . . . . . 8 |- B = U. K
12	11	toptopon 20722	. . . . . . 7 |- ( K e. Top <-> K e. (TopOn ` B ) )
13	10, 12	sylib 208	. . . . . 6 |- ( ph -> K e. (TopOn ` B ) )
14		cnflf 21806	. . . . . 6 |- ( ( ( J ↾t A ) e. (TopOn ` A ) /\ K e. (TopOn ` B ) ) -> ( F e. ( ( J ↾t A ) Cn K ) <-> ( F : A --> B /\ A. a e. ( Fil ` A ) A. x e. ( ( J ↾t A ) fLim a ) ( F ` x ) e. ( ( K fLimf a ) ` F ) ) ) )
15	8, 13, 14	syl2anc 693	. . . . 5 |- ( ph -> ( F e. ( ( J ↾t A ) Cn K ) <-> ( F : A --> B /\ A. a e. ( Fil ` A ) A. x e. ( ( J ↾t A ) fLim a ) ( F ` x ) e. ( ( K fLimf a ) ` F ) ) ) )
16	1, 15	mpbid 222	. . . 4 |- ( ph -> ( F : A --> B /\ A. a e. ( Fil ` A ) A. x e. ( ( J ↾t A ) fLim a ) ( F ` x ) e. ( ( K fLimf a ) ` F ) ) )
17	16	simprd 479	. . 3 |- ( ph -> A. a e. ( Fil ` A ) A. x e. ( ( J ↾t A ) fLim a ) ( F ` x ) e. ( ( K fLimf a ) ` F ) )
18	3	sscls 20860	. . . . . . 7 |- ( ( J e. Top /\ A C_ C ) -> A C_ ( ( cls ` J ) ` A ) )
19	2, 6, 18	syl2anc 693	. . . . . 6 |- ( ph -> A C_ ( ( cls ` J ) ` A ) )
20		flfcntr.y	. . . . . 6 |- ( ph -> X e. A )
21	19, 20	sseldd 3604	. . . . 5 |- ( ph -> X e. ( ( cls ` J ) ` A ) )
22	6, 20	sseldd 3604	. . . . . 6 |- ( ph -> X e. C )
23		trnei 21696	. . . . . 6 |- ( ( J e. (TopOn ` C ) /\ A C_ C /\ X e. C ) -> ( X e. ( ( cls ` J ) ` A ) <-> ( ( ( nei ` J ) ` { X } ) ↾t A ) e. ( Fil ` A ) ) )
24	5, 6, 22, 23	syl3anc 1326	. . . . 5 |- ( ph -> ( X e. ( ( cls ` J ) ` A ) <-> ( ( ( nei ` J ) ` { X } ) ↾t A ) e. ( Fil ` A ) ) )
25	21, 24	mpbid 222	. . . 4 |- ( ph -> ( ( ( nei ` J ) ` { X } ) ↾t A ) e. ( Fil ` A ) )
26		oveq2 6658	. . . . . 6 |- ( a = ( ( ( nei ` J ) ` { X } ) ↾t A ) -> ( ( J ↾t A ) fLim a ) = ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) )
27		oveq2 6658	. . . . . . . 8 |- ( a = ( ( ( nei ` J ) ` { X } ) ↾t A ) -> ( K fLimf a ) = ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) )
28	27	fveq1d 6193	. . . . . . 7 |- ( a = ( ( ( nei ` J ) ` { X } ) ↾t A ) -> ( ( K fLimf a ) ` F ) = ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) )
29	28	eleq2d 2687	. . . . . 6 |- ( a = ( ( ( nei ` J ) ` { X } ) ↾t A ) -> ( ( F ` x ) e. ( ( K fLimf a ) ` F ) <-> ( F ` x ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) ) )
30	26, 29	raleqbidv 3152	. . . . 5 |- ( a = ( ( ( nei ` J ) ` { X } ) ↾t A ) -> ( A. x e. ( ( J ↾t A ) fLim a ) ( F ` x ) e. ( ( K fLimf a ) ` F ) <-> A. x e. ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ( F ` x ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) ) )
31	30	adantl 482	. . . 4 |- ( ( ph /\ a = ( ( ( nei ` J ) ` { X } ) ↾t A ) ) -> ( A. x e. ( ( J ↾t A ) fLim a ) ( F ` x ) e. ( ( K fLimf a ) ` F ) <-> A. x e. ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ( F ` x ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) ) )
32	25, 31	rspcdv 3312	. . 3 |- ( ph -> ( A. a e. ( Fil ` A ) A. x e. ( ( J ↾t A ) fLim a ) ( F ` x ) e. ( ( K fLimf a ) ` F ) -> A. x e. ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ( F ` x ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) ) )
33	17, 32	mpd 15	. 2 |- ( ph -> A. x e. ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ( F ` x ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) )
34		neiflim 21778	. . . . 5 |- ( ( ( J ↾t A ) e. (TopOn ` A ) /\ X e. A ) -> X e. ( ( J ↾t A ) fLim ( ( nei ` ( J ↾t A ) ) ` { X } ) ) )
35	8, 20, 34	syl2anc 693	. . . 4 |- ( ph -> X e. ( ( J ↾t A ) fLim ( ( nei ` ( J ↾t A ) ) ` { X } ) ) )
36	20	snssd 4340	. . . . . 6 |- ( ph -> { X } C_ A )
37	3	neitr 20984	. . . . . 6 |- ( ( J e. Top /\ A C_ C /\ { X } C_ A ) -> ( ( nei ` ( J ↾t A ) ) ` { X } ) = ( ( ( nei ` J ) ` { X } ) ↾t A ) )
38	2, 6, 36, 37	syl3anc 1326	. . . . 5 |- ( ph -> ( ( nei ` ( J ↾t A ) ) ` { X } ) = ( ( ( nei ` J ) ` { X } ) ↾t A ) )
39	38	oveq2d 6666	. . . 4 |- ( ph -> ( ( J ↾t A ) fLim ( ( nei ` ( J ↾t A ) ) ` { X } ) ) = ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) )
40	35, 39	eleqtrd 2703	. . 3 |- ( ph -> X e. ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) )
41		fveq2 6191	. . . . 5 |- ( x = X -> ( F ` x ) = ( F ` X ) )
42	41	eleq1d 2686	. . . 4 |- ( x = X -> ( ( F ` x ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) <-> ( F ` X ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) ) )
43	42	adantl 482	. . 3 |- ( ( ph /\ x = X ) -> ( ( F ` x ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) <-> ( F ` X ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) ) )
44	40, 43	rspcdv 3312	. 2 |- ( ph -> ( A. x e. ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ( F ` x ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) -> ( F ` X ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) ) )
45	33, 44	mpd 15	1 |- ( ph -> ( F ` X ) e. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   {csn 4177   U.cuni 4436   -->wf 5884   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   clsccl 20822   neicnei 20901   Cn ccn 21028   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-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cn 21031  df-cnp 21032  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744

This theorem is referenced by:  cnextfres  21873