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Theorem flfcntr 24051
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 (𝜑𝐽 ∈ Top)
flfcntr.a (𝜑𝐴𝐶)
flfcntr.1 (𝜑𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾))
flfcntr.y (𝜑𝑋𝐴)
Assertion
Ref Expression
flfcntr (𝜑 → (𝐹𝑋) ∈ ((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.
StepHypRef Expression
1 fveq2 6906 . . 3 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
21eleq1d 2826 . 2 (𝑥 = 𝑋 → ((𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ↔ (𝐹𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
3 oveq2 7439 . . . 4 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐽t 𝐴) fLim 𝑎) = ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
4 oveq2 7439 . . . . . 6 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → (𝐾 fLimf 𝑎) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
54fveq1d 6908 . . . . 5 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐾 fLimf 𝑎)‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
65eleq2d 2827 . . . 4 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹) ↔ (𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
73, 6raleqbidv 3346 . . 3 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → (∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹) ↔ ∀𝑥 ∈ ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))(𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
8 flfcntr.1 . . . . 5 (𝜑𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾))
9 flfcntr.j . . . . . . . 8 (𝜑𝐽 ∈ Top)
10 flfcntr.c . . . . . . . . 9 𝐶 = 𝐽
1110toptopon 22923 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
129, 11sylib 218 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝐶))
13 flfcntr.a . . . . . . 7 (𝜑𝐴𝐶)
14 resttopon 23169 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
1512, 13, 14syl2anc 584 . . . . . 6 (𝜑 → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
16 cntop2 23249 . . . . . . . 8 (𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾) → 𝐾 ∈ Top)
178, 16syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
18 flfcntr.b . . . . . . . 8 𝐵 = 𝐾
1918toptopon 22923 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
2017, 19sylib 218 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘𝐵))
21 cnflf 24010 . . . . . 6 (((𝐽t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘𝐵)) → (𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))))
2215, 20, 21syl2anc 584 . . . . 5 (𝜑 → (𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))))
238, 22mpbid 232 . . . 4 (𝜑 → (𝐹:𝐴𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹)))
2423simprd 495 . . 3 (𝜑 → ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))
2510sscls 23064 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
269, 13, 25syl2anc 584 . . . . 5 (𝜑𝐴 ⊆ ((cls‘𝐽)‘𝐴))
27 flfcntr.y . . . . 5 (𝜑𝑋𝐴)
2826, 27sseldd 3984 . . . 4 (𝜑𝑋 ∈ ((cls‘𝐽)‘𝐴))
2913, 27sseldd 3984 . . . . 5 (𝜑𝑋𝐶)
30 trnei 23900 . . . . 5 ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶𝑋𝐶) → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3112, 13, 29, 30syl3anc 1373 . . . 4 (𝜑 → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3228, 31mpbid 232 . . 3 (𝜑 → (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴))
337, 24, 32rspcdva 3623 . 2 (𝜑 → ∀𝑥 ∈ ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))(𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
34 neiflim 23982 . . . 4 (((𝐽t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝑋𝐴) → 𝑋 ∈ ((𝐽t 𝐴) fLim ((nei‘(𝐽t 𝐴))‘{𝑋})))
3515, 27, 34syl2anc 584 . . 3 (𝜑𝑋 ∈ ((𝐽t 𝐴) fLim ((nei‘(𝐽t 𝐴))‘{𝑋})))
3627snssd 4809 . . . . 5 (𝜑 → {𝑋} ⊆ 𝐴)
3710neitr 23188 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝐶 ∧ {𝑋} ⊆ 𝐴) → ((nei‘(𝐽t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
389, 13, 36, 37syl3anc 1373 . . . 4 (𝜑 → ((nei‘(𝐽t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
3938oveq2d 7447 . . 3 (𝜑 → ((𝐽t 𝐴) fLim ((nei‘(𝐽t 𝐴))‘{𝑋})) = ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
4035, 39eleqtrd 2843 . 2 (𝜑𝑋 ∈ ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
412, 33, 40rspcdva 3623 1 (𝜑 → (𝐹𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wss 3951  {csn 4626   cuni 4907  wf 6557  cfv 6561  (class class class)co 7431  t crest 17465  Topctop 22899  TopOnctopon 22916  clsccl 23026  neicnei 23105   Cn ccn 23232  Filcfil 23853   fLim cflim 23942   fLimf cflf 23943
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-map 8868  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-fbas 21361  df-fg 21362  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-cn 23235  df-cnp 23236  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948
This theorem is referenced by:  cnextfres  24077
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