Theorem flfcntr 23991

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 6835	. . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
2	1	eleq1d 2822	. 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 7368	. . . 4 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐽 ↾t 𝐴) fLim 𝑎) = ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
4		oveq2 7368	. . . . . 6 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → (𝐾 fLimf 𝑎) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
5	4	fveq1d 6837	. . . . 5 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐾 fLimf 𝑎)‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
6	5	eleq2d 2823	. . . 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 3317	. . 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 22865	. . . . . . . 8 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
12	9, 11	sylib 218	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐶))
13		flfcntr.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
14		resttopon 23109	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
15	12, 13, 14	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
16		cntop2 23189	. . . . . . . 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 22865	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
20	17, 19	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐵))
21		cnflf 23950	. . . . . 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 585	. . . . 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 232	. . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim 𝑎)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹)))
24	23	simprd 495	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim 𝑎)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))
25	10	sscls 23004	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
26	9, 13, 25	syl2anc 585	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
27		flfcntr.y	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
28	26, 27	sseldd 3935	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((cls‘𝐽)‘𝐴))
29	13, 27	sseldd 3935	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐶)
30		trnei 23840	. . . . 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 1374	. . . 4 ⊢ (𝜑 → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴)))
32	28, 31	mpbid 232	. . 3 ⊢ (𝜑 → (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴))
33	7, 24, 32	rspcdva 3578	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))(𝐹‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
34		neiflim 23922	. . . 4 ⊢ (((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋})))
35	15, 27, 34	syl2anc 585	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋})))
36	27	snssd 4766	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝐴)
37	10	neitr 23128	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶 ∧ {𝑋} ⊆ 𝐴) → ((nei‘(𝐽 ↾t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
38	9, 13, 36, 37	syl3anc 1374	. . . 4 ⊢ (𝜑 → ((nei‘(𝐽 ↾t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
39	38	oveq2d 7376	. . 3 ⊢ (𝜑 → ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋})) = ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
40	35, 39	eleqtrd 2839	. 2 ⊢ (𝜑 → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
41	2, 33, 40	rspcdva 3578	1 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3902  {csn 4581  ∪ cuni 4864  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360   ↾_t crest 17344  Topctop 22841  TopOnctopon 22858  clsccl 22966  neicnei 23045   Cn ccn 23172  Filcfil 23793   fLim cflim 23882   fLimf cflf 23883

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-map 8769  df-en 8888  df-fin 8891  df-fi 9318  df-rest 17346  df-topgen 17367  df-fbas 21310  df-fg 21311  df-top 22842  df-topon 22859  df-bases 22894  df-cld 22967  df-ntr 22968  df-cls 22969  df-nei 23046  df-cn 23175  df-cnp 23176  df-fil 23794  df-fm 23886  df-flim 23887  df-flf 23888

This theorem is referenced by:  cnextfres  24017