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Theorem flfcntr 22067
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 6332 . . 3 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
21eleq1d 2835 . 2 (𝑥 = 𝑋 → ((𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ↔ (𝐹𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
3 oveq2 6801 . . . 4 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐽t 𝐴) fLim 𝑎) = ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
4 oveq2 6801 . . . . . 6 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → (𝐾 fLimf 𝑎) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
54fveq1d 6334 . . . . 5 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐾 fLimf 𝑎)‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
65eleq2d 2836 . . . 4 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹) ↔ (𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
73, 6raleqbidv 3301 . . 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 20942 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
129, 11sylib 208 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝐶))
13 flfcntr.a . . . . . . 7 (𝜑𝐴𝐶)
14 resttopon 21186 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
1512, 13, 14syl2anc 573 . . . . . 6 (𝜑 → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
16 cntop2 21266 . . . . . . . 8 (𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾) → 𝐾 ∈ Top)
178, 16syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
18 flfcntr.b . . . . . . . 8 𝐵 = 𝐾
1918toptopon 20942 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
2017, 19sylib 208 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘𝐵))
21 cnflf 22026 . . . . . 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 573 . . . . 5 (𝜑 → (𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))))
238, 22mpbid 222 . . . 4 (𝜑 → (𝐹:𝐴𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹)))
2423simprd 483 . . 3 (𝜑 → ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))
2510sscls 21081 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
269, 13, 25syl2anc 573 . . . . 5 (𝜑𝐴 ⊆ ((cls‘𝐽)‘𝐴))
27 flfcntr.y . . . . 5 (𝜑𝑋𝐴)
2826, 27sseldd 3753 . . . 4 (𝜑𝑋 ∈ ((cls‘𝐽)‘𝐴))
2913, 27sseldd 3753 . . . . 5 (𝜑𝑋𝐶)
30 trnei 21916 . . . . 5 ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶𝑋𝐶) → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3112, 13, 29, 30syl3anc 1476 . . . 4 (𝜑 → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3228, 31mpbid 222 . . 3 (𝜑 → (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴))
337, 24, 32rspcdva 3466 . 2 (𝜑 → ∀𝑥 ∈ ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))(𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
34 neiflim 21998 . . . 4 (((𝐽t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝑋𝐴) → 𝑋 ∈ ((𝐽t 𝐴) fLim ((nei‘(𝐽t 𝐴))‘{𝑋})))
3515, 27, 34syl2anc 573 . . 3 (𝜑𝑋 ∈ ((𝐽t 𝐴) fLim ((nei‘(𝐽t 𝐴))‘{𝑋})))
3627snssd 4475 . . . . 5 (𝜑 → {𝑋} ⊆ 𝐴)
3710neitr 21205 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝐶 ∧ {𝑋} ⊆ 𝐴) → ((nei‘(𝐽t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
389, 13, 36, 37syl3anc 1476 . . . 4 (𝜑 → ((nei‘(𝐽t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
3938oveq2d 6809 . . 3 (𝜑 → ((𝐽t 𝐴) fLim ((nei‘(𝐽t 𝐴))‘{𝑋})) = ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
4035, 39eleqtrd 2852 . 2 (𝜑𝑋 ∈ ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
412, 33, 40rspcdva 3466 1 (𝜑 → (𝐹𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wss 3723  {csn 4316   cuni 4574  wf 6027  cfv 6031  (class class class)co 6793  t crest 16289  Topctop 20918  TopOnctopon 20935  clsccl 21043  neicnei 21122   Cn ccn 21249  Filcfil 21869   fLim cflim 21958   fLimf cflf 21959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-fin 8113  df-fi 8473  df-rest 16291  df-topgen 16312  df-fbas 19958  df-fg 19959  df-top 20919  df-topon 20936  df-bases 20971  df-cld 21044  df-ntr 21045  df-cls 21046  df-nei 21123  df-cn 21252  df-cnp 21253  df-fil 21870  df-fm 21962  df-flim 21963  df-flf 21964
This theorem is referenced by:  cnextfres  22093
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