Theorem flfcntr 22648

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 = ∪ 𝐽
flfcntr.b	⊢ 𝐵 = ∪ 𝐾
flfcntr.j	⊢ (𝜑 → 𝐽 ∈ Top)
flfcntr.a	⊢ (𝜑 → 𝐴 ⊆ 𝐶)
flfcntr.1	⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))
flfcntr.y	⊢ (𝜑 → 𝑋 ∈ 𝐴)

Assertion

Ref	Expression
flfcntr	⊢ (𝜑 → (𝐹‘𝑋) ∈ ((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		fveq2 6645	. . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
2	1	eleq1d 2874	. 2 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ↔ (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
3		oveq2 7143	. . . 4 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐽 ↾t 𝐴) fLim 𝑎) = ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
4		oveq2 7143	. . . . . 6 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → (𝐾 fLimf 𝑎) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
5	4	fveq1d 6647	. . . . 5 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐾 fLimf 𝑎)‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
6	5	eleq2d 2875	. . . 4 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹) ↔ (𝐹‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
7	3, 6	raleqbidv 3354	. . 3 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → (∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim 𝑎)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹) ↔ ∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))(𝐹‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
8		flfcntr.1	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))
9		flfcntr.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Top)
10		flfcntr.c	. . . . . . . . 9 ⊢ 𝐶 = ∪ 𝐽
11	10	toptopon 21522	. . . . . . . 8 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
12	9, 11	sylib 221	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐶))
13		flfcntr.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
14		resttopon 21766	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
15	12, 13, 14	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
16		cntop2 21846	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) → 𝐾 ∈ Top)
17	8, 16	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
18		flfcntr.b	. . . . . . . 8 ⊢ 𝐵 = ∪ 𝐾
19	18	toptopon 21522	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
20	17, 19	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐵))
21		cnflf 22607	. . . . . 6 ⊢ (((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘𝐵)) → (𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim 𝑎)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))))
22	15, 20, 21	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim 𝑎)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))))
23	8, 22	mpbid 235	. . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim 𝑎)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹)))
24	23	simprd 499	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim 𝑎)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))
25	10	sscls 21661	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
26	9, 13, 25	syl2anc 587	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
27		flfcntr.y	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
28	26, 27	sseldd 3916	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((cls‘𝐽)‘𝐴))
29	13, 27	sseldd 3916	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐶)
30		trnei 22497	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑋 ∈ 𝐶) → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴)))
31	12, 13, 29, 30	syl3anc 1368	. . . 4 ⊢ (𝜑 → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴)))
32	28, 31	mpbid 235	. . 3 ⊢ (𝜑 → (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴))
33	7, 24, 32	rspcdva 3573	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))(𝐹‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
34		neiflim 22579	. . . 4 ⊢ (((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋})))
35	15, 27, 34	syl2anc 587	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋})))
36	27	snssd 4702	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝐴)
37	10	neitr 21785	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶 ∧ {𝑋} ⊆ 𝐴) → ((nei‘(𝐽 ↾t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
38	9, 13, 36, 37	syl3anc 1368	. . . 4 ⊢ (𝜑 → ((nei‘(𝐽 ↾t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
39	38	oveq2d 7151	. . 3 ⊢ (𝜑 → ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋})) = ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
40	35, 39	eleqtrd 2892	. 2 ⊢ (𝜑 → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
41	2, 33, 40	rspcdva 3573	1 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881  {csn 4525  ∪ cuni 4800  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ↾_t crest 16686  Topctop 21498  TopOnctopon 21515  clsccl 21623  neicnei 21702   Cn ccn 21829  Filcfil 22450   fLim cflim 22539   fLimf cflf 22540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-fin 8496  df-fi 8859  df-rest 16688  df-topgen 16709  df-fbas 20088  df-fg 20089  df-top 21499  df-topon 21516  df-bases 21551  df-cld 21624  df-ntr 21625  df-cls 21626  df-nei 21703  df-cn 21832  df-cnp 21833  df-fil 22451  df-fm 22543  df-flim 22544  df-flf 22545

This theorem is referenced by:  cnextfres  22674