Theorem flfcntr 21136

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 20025	. . . . . . . 8 |- ( J e. Top <-> J e. (TopOn ` C ) )
5	2, 4	sylib 201	. . . . . . 7 |- ( ph -> J e. (TopOn ` C ) )
6		flfcntr.a	. . . . . . 7 |- ( ph -> A C_ C )
7		resttopon 20254	. . . . . . 7 |- ( ( J e. (TopOn ` C ) /\ A C_ C ) -> ( J ↾t A ) e. (TopOn ` A ) )
8	5, 6, 7	syl2anc 673	. . . . . 6 |- ( ph -> ( J ↾t A ) e. (TopOn ` A ) )
9		cntop2 20334	. . . . . . . 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 20025	. . . . . . 7 |- ( K e. Top <-> K e. (TopOn ` B ) )
13	10, 12	sylib 201	. . . . . 6 |- ( ph -> K e. (TopOn ` B ) )
14		cnflf 21095	. . . . . 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 673	. . . . 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 215	. . . 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 470	. . 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 20148	. . . . . . 7 |- ( ( J e. Top /\ A C_ C ) -> A C_ ( ( cls ` J ) ` A ) )
19	2, 6, 18	syl2anc 673	. . . . . 6 |- ( ph -> A C_ ( ( cls ` J ) ` A ) )
20		flfcntr.y	. . . . . 6 |- ( ph -> X e. A )
21	19, 20	sseldd 3419	. . . . 5 |- ( ph -> X e. ( ( cls ` J ) ` A ) )
22	6, 20	sseldd 3419	. . . . . 6 |- ( ph -> X e. C )
23		trnei 20985	. . . . . 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 1292	. . . . 5 |- ( ph -> ( X e. ( ( cls ` J ) ` A ) <-> ( ( ( nei ` J ) ` { X } ) ↾t A ) e. ( Fil ` A ) ) )
25	21, 24	mpbid 215	. . . 4 |- ( ph -> ( ( ( nei ` J ) ` { X } ) ↾t A ) e. ( Fil ` A ) )
26		oveq2 6316	. . . . . 6 |- ( a = ( ( ( nei ` J ) ` { X } ) ↾t A ) -> ( ( J ↾t A ) fLim a ) = ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) )
27		oveq2 6316	. . . . . . . 8 |- ( a = ( ( ( nei ` J ) ` { X } ) ↾t A ) -> ( K fLimf a ) = ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) )
28	27	fveq1d 5881	. . . . . . 7 |- ( a = ( ( ( nei ` J ) ` { X } ) ↾t A ) -> ( ( K fLimf a ) ` F ) = ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) )
29	28	eleq2d 2534	. . . . . 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 2987	. . . . 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 473	. . . 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 3139	. . 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 21067	. . . . 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 673	. . . 4 |- ( ph -> X e. ( ( J ↾t A ) fLim ( ( nei ` ( J ↾t A ) ) ` { X } ) ) )
36	20	snssd 4108	. . . . . 6 |- ( ph -> { X } C_ A )
37	3	neitr 20273	. . . . . 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 1292	. . . . 5 |- ( ph -> ( ( nei ` ( J ↾t A ) ) ` { X } ) = ( ( ( nei ` J ) ` { X } ) ↾t A ) )
39	38	oveq2d 6324	. . . 4 |- ( ph -> ( ( J ↾t A ) fLim ( ( nei ` ( J ↾t A ) ) ` { X } ) ) = ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) )
40	35, 39	eleqtrd 2551	. . 3 |- ( ph -> X e. ( ( J ↾t A ) fLim ( ( ( nei ` J ) ` { X } ) ↾t A ) ) )
41		fveq2 5879	. . . . 5 |- ( x = X -> ( F ` x ) = ( F ` X ) )
42	41	eleq1d 2533	. . . 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 473	. . 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 3139	. 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 189   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   C_ wss 3390   {csn 3959   U.cuni 4190   -->wf 5585   ` cfv 5589  (class class class)co 6308   ↾_t crest 15397   Topctop 19994  TopOnctopon 19995   clsccl 20110   neicnei 20190   Cn ccn 20317   Filcfil 20938   fLim cflim 21027   fLimf cflf 21028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-fin 7591  df-fi 7943  df-rest 15399  df-topgen 15420  df-fbas 19044  df-fg 19045  df-top 19998  df-bases 19999  df-topon 20000  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-cn 20320  df-cnp 20321  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033

This theorem is referenced by:  cnextfres  21162