Theorem flfcntr 23937

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 6861	. . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
2	1	eleq1d 2814	. 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 7398	. . . 4 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐽 ↾t 𝐴) fLim 𝑎) = ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
4		oveq2 7398	. . . . . 6 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → (𝐾 fLimf 𝑎) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
5	4	fveq1d 6863	. . . . 5 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐾 fLimf 𝑎)‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
6	5	eleq2d 2815	. . . 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 3321	. . 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 22811	. . . . . . . 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 23055	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
15	12, 13, 14	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
16		cntop2 23135	. . . . . . . 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 22811	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
20	17, 19	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐵))
21		cnflf 23896	. . . . . 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 584	. . . . 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 22950	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
26	9, 13, 25	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
27		flfcntr.y	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
28	26, 27	sseldd 3950	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((cls‘𝐽)‘𝐴))
29	13, 27	sseldd 3950	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐶)
30		trnei 23786	. . . . 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 1373	. . . 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 3592	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))(𝐹‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
34		neiflim 23868	. . . 4 ⊢ (((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋})))
35	15, 27, 34	syl2anc 584	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋})))
36	27	snssd 4776	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝐴)
37	10	neitr 23074	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶 ∧ {𝑋} ⊆ 𝐴) → ((nei‘(𝐽 ↾t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
38	9, 13, 36, 37	syl3anc 1373	. . . 4 ⊢ (𝜑 → ((nei‘(𝐽 ↾t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
39	38	oveq2d 7406	. . 3 ⊢ (𝜑 → ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋})) = ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
40	35, 39	eleqtrd 2831	. 2 ⊢ (𝜑 → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
41	2, 33, 40	rspcdva 3592	1 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ⊆ wss 3917  {csn 4592  ∪ cuni 4874  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ↾_t crest 17390  Topctop 22787  TopOnctopon 22804  clsccl 22912  neicnei 22991   Cn ccn 23118  Filcfil 23739   fLim cflim 23828   fLimf cflf 23829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-map 8804  df-en 8922  df-fin 8925  df-fi 9369  df-rest 17392  df-topgen 17413  df-fbas 21268  df-fg 21269  df-top 22788  df-topon 22805  df-bases 22840  df-cld 22913  df-ntr 22914  df-cls 22915  df-nei 22992  df-cn 23121  df-cnp 23122  df-fil 23740  df-fm 23832  df-flim 23833  df-flf 23834

This theorem is referenced by:  cnextfres  23963