Theorem flfcntr 24072

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 6920	. . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
2	1	eleq1d 2829	. 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 7456	. . . 4 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐽 ↾t 𝐴) fLim 𝑎) = ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
4		oveq2 7456	. . . . . 6 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → (𝐾 fLimf 𝑎) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
5	4	fveq1d 6922	. . . . 5 ⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐾 fLimf 𝑎)‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
6	5	eleq2d 2830	. . . 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 22944	. . . . . . . 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 23190	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
15	12, 13, 14	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
16		cntop2 23270	. . . . . . . 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 22944	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
20	17, 19	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐵))
21		cnflf 24031	. . . . . 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 583	. . . . 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 23085	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
26	9, 13, 25	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
27		flfcntr.y	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
28	26, 27	sseldd 4009	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((cls‘𝐽)‘𝐴))
29	13, 27	sseldd 4009	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐶)
30		trnei 23921	. . . . 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 1371	. . . 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 3636	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))(𝐹‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
34		neiflim 24003	. . . 4 ⊢ (((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋})))
35	15, 27, 34	syl2anc 583	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋})))
36	27	snssd 4834	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝐴)
37	10	neitr 23209	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶 ∧ {𝑋} ⊆ 𝐴) → ((nei‘(𝐽 ↾t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
38	9, 13, 36, 37	syl3anc 1371	. . . 4 ⊢ (𝜑 → ((nei‘(𝐽 ↾t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
39	38	oveq2d 7464	. . 3 ⊢ (𝜑 → ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋})) = ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
40	35, 39	eleqtrd 2846	. 2 ⊢ (𝜑 → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
41	2, 33, 40	rspcdva 3636	1 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  {csn 4648  ∪ cuni 4931  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↾_t crest 17480  Topctop 22920  TopOnctopon 22937  clsccl 23047  neicnei 23126   Cn ccn 23253  Filcfil 23874   fLim cflim 23963   fLimf cflf 23964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-map 8886  df-en 9004  df-fin 9007  df-fi 9480  df-rest 17482  df-topgen 17503  df-fbas 21384  df-fg 21385  df-top 22921  df-topon 22938  df-bases 22974  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-cn 23256  df-cnp 23257  df-fil 23875  df-fm 23967  df-flim 23968  df-flf 23969

This theorem is referenced by:  cnextfres  24098