Theorem flfcntr 21058

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 19948	. . . . . . . 8 |- ( J e. Top <-> J e. (TopOn ` C ) )
5	2, 4	sylib 200	. . . . . . 7 |- ( ph -> J e. (TopOn ` C ) )
6		flfcntr.a	. . . . . . 7 |- ( ph -> A C_ C )
7		resttopon 20177	. . . . . . 7 |- ( ( J e. (TopOn ` C ) /\ A C_ C ) -> ( J ↾t A ) e. (TopOn ` A ) )
8	5, 6, 7	syl2anc 667	. . . . . 6 |- ( ph -> ( J ↾t A ) e. (TopOn ` A ) )
9		cntop2 20257	. . . . . . . 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 19948	. . . . . . 7 |- ( K e. Top <-> K e. (TopOn ` B ) )
13	10, 12	sylib 200	. . . . . 6 |- ( ph -> K e. (TopOn ` B ) )
14		cnflf 21017	. . . . . 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 667	. . . . 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 214	. . . 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 465	. . 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 20071	. . . . . . 7 |- ( ( J e. Top /\ A C_ C ) -> A C_ ( ( cls ` J ) ` A ) )
19	2, 6, 18	syl2anc 667	. . . . . 6 |- ( ph -> A C_ ( ( cls ` J ) ` A ) )
20		flfcntr.y	. . . . . 6 |- ( ph -> X e. A )
21	19, 20	sseldd 3433	. . . . 5 |- ( ph -> X e. ( ( cls ` J ) ` A ) )
22	6, 20	sseldd 3433	. . . . . 6 |- ( ph -> X e. C )
23		trnei 20907	. . . . . 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 1268	. . . . 5 |- ( ph -> ( X e. ( ( cls ` J ) ` A ) <-> ( ( ( nei ` J ) ` { X } ) ↾t A ) e. ( Fil ` A ) ) )
25	21, 24	mpbid 214	. . . 4 |- ( ph -> ( ( ( nei ` J ) ` { X } ) ↾t A ) e. ( Fil ` A ) )
26		oveq2 6298	. . . . . 6 |- ( a = ( ( ( nei ` J ) ` { X } ) ↾t A ) -> ( ( J ↾t A ) fLim a ) = ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) )
27		oveq2 6298	. . . . . . . 8 |- ( a = ( ( ( nei ` J ) ` { X } ) ↾t A ) -> ( K fLimf a ) = ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) )
28	27	fveq1d 5867	. . . . . . 7 |- ( a = ( ( ( nei ` J ) ` { X } ) ↾t A ) -> ( ( K fLimf a ) ` F ) = ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) )
29	28	eleq2d 2514	. . . . . 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 3001	. . . . 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 468	. . . 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 3153	. . 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 20989	. . . . 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 667	. . . 4 |- ( ph -> X e. ( ( J ↾t A ) fLim ( ( nei ` ( J ↾t A ) ) ` { X } ) ) )
36	20	snssd 4117	. . . . . 6 |- ( ph -> { X } C_ A )
37	3	neitr 20196	. . . . . 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 1268	. . . . 5 |- ( ph -> ( ( nei ` ( J ↾t A ) ) ` { X } ) = ( ( ( nei ` J ) ` { X } ) ↾t A ) )
39	38	oveq2d 6306	. . . 4 |- ( ph -> ( ( J ↾t A ) fLim ( ( nei ` ( J ↾t A ) ) ` { X } ) ) = ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) )
40	35, 39	eleqtrd 2531	. . 3 |- ( ph -> X e. ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) )
41		fveq2 5865	. . . . 5 |- ( x = X -> ( F ` x ) = ( F ` X ) )
42	41	eleq1d 2513	. . . 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 468	. . 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 3153	. 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 188   /\ wa 371   = wceq 1444   e. wcel 1887   A.wral 2737   C_ wss 3404   {csn 3968   U.cuni 4198   -->wf 5578   ` cfv 5582  (class class class)co 6290   ↾_t crest 15319   Topctop 19917  TopOnctopon 19918   clsccl 20033   neicnei 20113   Cn ccn 20240   Filcfil 20860   fLim cflim 20949   fLimf cflf 20950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-fin 7573  df-fi 7925  df-rest 15321  df-topgen 15342  df-fbas 18967  df-fg 18968  df-top 19921  df-bases 19922  df-topon 19923  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-cn 20243  df-cnp 20244  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955

This theorem is referenced by:  cnextfres  21084