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Theorem flfcntr 22064
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 6411 . . 3 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
21eleq1d 2877 . 2 (𝑥 = 𝑋 → ((𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ↔ (𝐹𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
3 oveq2 6885 . . . 4 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐽t 𝐴) fLim 𝑎) = ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
4 oveq2 6885 . . . . . 6 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → (𝐾 fLimf 𝑎) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
54fveq1d 6413 . . . . 5 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐾 fLimf 𝑎)‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
65eleq2d 2878 . . . 4 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹) ↔ (𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
73, 6raleqbidv 3348 . . 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 20939 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
129, 11sylib 209 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝐶))
13 flfcntr.a . . . . . . 7 (𝜑𝐴𝐶)
14 resttopon 21183 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
1512, 13, 14syl2anc 575 . . . . . 6 (𝜑 → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
16 cntop2 21263 . . . . . . . 8 (𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾) → 𝐾 ∈ Top)
178, 16syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
18 flfcntr.b . . . . . . . 8 𝐵 = 𝐾
1918toptopon 20939 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
2017, 19sylib 209 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘𝐵))
21 cnflf 22023 . . . . . 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 575 . . . . 5 (𝜑 → (𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))))
238, 22mpbid 223 . . . 4 (𝜑 → (𝐹:𝐴𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹)))
2423simprd 485 . . 3 (𝜑 → ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))
2510sscls 21078 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
269, 13, 25syl2anc 575 . . . . 5 (𝜑𝐴 ⊆ ((cls‘𝐽)‘𝐴))
27 flfcntr.y . . . . 5 (𝜑𝑋𝐴)
2826, 27sseldd 3806 . . . 4 (𝜑𝑋 ∈ ((cls‘𝐽)‘𝐴))
2913, 27sseldd 3806 . . . . 5 (𝜑𝑋𝐶)
30 trnei 21913 . . . . 5 ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶𝑋𝐶) → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3112, 13, 29, 30syl3anc 1483 . . . 4 (𝜑 → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴)))
3228, 31mpbid 223 . . 3 (𝜑 → (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴))
337, 24, 32rspcdva 3515 . 2 (𝜑 → ∀𝑥 ∈ ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))(𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
34 neiflim 21995 . . . 4 (((𝐽t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝑋𝐴) → 𝑋 ∈ ((𝐽t 𝐴) fLim ((nei‘(𝐽t 𝐴))‘{𝑋})))
3515, 27, 34syl2anc 575 . . 3 (𝜑𝑋 ∈ ((𝐽t 𝐴) fLim ((nei‘(𝐽t 𝐴))‘{𝑋})))
3627snssd 4537 . . . . 5 (𝜑 → {𝑋} ⊆ 𝐴)
3710neitr 21202 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝐶 ∧ {𝑋} ⊆ 𝐴) → ((nei‘(𝐽t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
389, 13, 36, 37syl3anc 1483 . . . 4 (𝜑 → ((nei‘(𝐽t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
3938oveq2d 6893 . . 3 (𝜑 → ((𝐽t 𝐴) fLim ((nei‘(𝐽t 𝐴))‘{𝑋})) = ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
4035, 39eleqtrd 2894 . 2 (𝜑𝑋 ∈ ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
412, 33, 40rspcdva 3515 1 (𝜑 → (𝐹𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  wral 3103  wss 3776  {csn 4377   cuni 4637  wf 6100  cfv 6104  (class class class)co 6877  t crest 16289  Topctop 20915  TopOnctopon 20932  clsccl 21040  neicnei 21119   Cn ccn 21246  Filcfil 21866   fLim cflim 21955   fLimf cflf 21956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-iin 4722  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-oadd 7803  df-er 7982  df-map 8097  df-en 8196  df-fin 8199  df-fi 8559  df-rest 16291  df-topgen 16312  df-fbas 19954  df-fg 19955  df-top 20916  df-topon 20933  df-bases 20968  df-cld 21041  df-ntr 21042  df-cls 21043  df-nei 21120  df-cn 21249  df-cnp 21250  df-fil 21867  df-fm 21959  df-flim 21960  df-flf 21961
This theorem is referenced by:  cnextfres  22090
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