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Theorem flfcntr 24067
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 6907 . . 3 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
21eleq1d 2824 . 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 6909 . . . . 5 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐾 fLimf 𝑎)‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
65eleq2d 2825 . . . 4 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹) ↔ (𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
73, 6raleqbidv 3344 . . 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 22939 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
129, 11sylib 218 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝐶))
13 flfcntr.a . . . . . . 7 (𝜑𝐴𝐶)
14 resttopon 23185 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
1512, 13, 14syl2anc 584 . . . . . 6 (𝜑 → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
16 cntop2 23265 . . . . . . . 8 (𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾) → 𝐾 ∈ Top)
178, 16syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
18 flfcntr.b . . . . . . . 8 𝐵 = 𝐾
1918toptopon 22939 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
2017, 19sylib 218 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘𝐵))
21 cnflf 24026 . . . . . 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 23080 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
269, 13, 25syl2anc 584 . . . . 5 (𝜑𝐴 ⊆ ((cls‘𝐽)‘𝐴))
27 flfcntr.y . . . . 5 (𝜑𝑋𝐴)
2826, 27sseldd 3996 . . . 4 (𝜑𝑋 ∈ ((cls‘𝐽)‘𝐴))
2913, 27sseldd 3996 . . . . 5 (𝜑𝑋𝐶)
30 trnei 23916 . . . . 5 ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶𝑋𝐶) → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3112, 13, 29, 30syl3anc 1370 . . . 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 23998 . . . 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 4814 . . . . 5 (𝜑 → {𝑋} ⊆ 𝐴)
3710neitr 23204 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝐶 ∧ {𝑋} ⊆ 𝐴) → ((nei‘(𝐽t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
389, 13, 36, 37syl3anc 1370 . . . 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 2841 . 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 1537  wcel 2106  wral 3059  wss 3963  {csn 4631   cuni 4912  wf 6559  cfv 6563  (class class class)co 7431  t crest 17467  Topctop 22915  TopOnctopon 22932  clsccl 23042  neicnei 23121   Cn ccn 23248  Filcfil 23869   fLim cflim 23958   fLimf cflf 23959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-map 8867  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-cn 23251  df-cnp 23252  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964
This theorem is referenced by:  cnextfres  24093
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