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Theorem flfcntr 21757
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 flfcntr.1 . . . . 5 (𝜑𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾))
2 flfcntr.j . . . . . . . 8 (𝜑𝐽 ∈ Top)
3 flfcntr.c . . . . . . . . 9 𝐶 = 𝐽
43toptopon 20648 . . . . . . . 8 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
52, 4sylib 208 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝐶))
6 flfcntr.a . . . . . . 7 (𝜑𝐴𝐶)
7 resttopon 20875 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
85, 6, 7syl2anc 692 . . . . . 6 (𝜑 → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
9 cntop2 20955 . . . . . . . 8 (𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾) → 𝐾 ∈ Top)
101, 9syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
11 flfcntr.b . . . . . . . 8 𝐵 = 𝐾
1211toptopon 20648 . . . . . . 7 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
1310, 12sylib 208 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘𝐵))
14 cnflf 21716 . . . . . 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))))
158, 13, 14syl2anc 692 . . . . 5 (𝜑 → (𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))))
161, 15mpbid 222 . . . 4 (𝜑 → (𝐹:𝐴𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹)))
1716simprd 479 . . 3 (𝜑 → ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽t 𝐴) fLim 𝑎)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))
183sscls 20770 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
192, 6, 18syl2anc 692 . . . . . 6 (𝜑𝐴 ⊆ ((cls‘𝐽)‘𝐴))
20 flfcntr.y . . . . . 6 (𝜑𝑋𝐴)
2119, 20sseldd 3584 . . . . 5 (𝜑𝑋 ∈ ((cls‘𝐽)‘𝐴))
226, 20sseldd 3584 . . . . . 6 (𝜑𝑋𝐶)
23 trnei 21606 . . . . . 6 ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴𝐶𝑋𝐶) → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴)))
245, 6, 22, 23syl3anc 1323 . . . . 5 (𝜑 → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴)))
2521, 24mpbid 222 . . . 4 (𝜑 → (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴))
26 oveq2 6612 . . . . . 6 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐽t 𝐴) fLim 𝑎) = ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
27 oveq2 6612 . . . . . . . 8 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → (𝐾 fLimf 𝑎) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
2827fveq1d 6150 . . . . . . 7 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐾 fLimf 𝑎)‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
2928eleq2d 2684 . . . . . 6 (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐹𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹) ↔ (𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
3026, 29raleqbidv 3141 . . . . 5 (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)))
3130adantl 482 . . . 4 ((𝜑𝑎 = (((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)))
3225, 31rspcdv 3298 . . 3 (𝜑 → (∀𝑎 ∈ (Fil‘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)))
3317, 32mpd 15 . 2 (𝜑 → ∀𝑥 ∈ ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))(𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
34 neiflim 21688 . . . . 5 (((𝐽t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝑋𝐴) → 𝑋 ∈ ((𝐽t 𝐴) fLim ((nei‘(𝐽t 𝐴))‘{𝑋})))
358, 20, 34syl2anc 692 . . . 4 (𝜑𝑋 ∈ ((𝐽t 𝐴) fLim ((nei‘(𝐽t 𝐴))‘{𝑋})))
3620snssd 4309 . . . . . 6 (𝜑 → {𝑋} ⊆ 𝐴)
373neitr 20894 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝐶 ∧ {𝑋} ⊆ 𝐴) → ((nei‘(𝐽t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
382, 6, 36, 37syl3anc 1323 . . . . 5 (𝜑 → ((nei‘(𝐽t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))
3938oveq2d 6620 . . . 4 (𝜑 → ((𝐽t 𝐴) fLim ((nei‘(𝐽t 𝐴))‘{𝑋})) = ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
4035, 39eleqtrd 2700 . . 3 (𝜑𝑋 ∈ ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)))
41 fveq2 6148 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
4241eleq1d 2683 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ↔ (𝐹𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
4342adantl 482 . . 3 ((𝜑𝑥 = 𝑋) → ((𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ↔ (𝐹𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
4440, 43rspcdv 3298 . 2 (𝜑 → (∀𝑥 ∈ ((𝐽t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))(𝐹𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) → (𝐹𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)))
4533, 44mpd 15 1 (𝜑 → (𝐹𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3555  {csn 4148   cuni 4402  wf 5843  cfv 5847  (class class class)co 6604  t crest 16002  Topctop 20617  TopOnctopon 20618  clsccl 20732  neicnei 20811   Cn ccn 20938  Filcfil 21559   fLim cflim 21648   fLimf cflf 21649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-fbas 19662  df-fg 19663  df-top 20621  df-bases 20622  df-topon 20623  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-cn 20941  df-cnp 20942  df-fil 21560  df-fm 21652  df-flim 21653  df-flf 21654
This theorem is referenced by:  cnextfres  21783
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