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Theorem cnpfcfi 24064
Description: Lemma for cnpfcf 24065. If a function is continuous at a point, it respects clustering there. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
cnpfcfi ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝐿)‘𝐹))

Proof of Theorem cnpfcfi
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 simp2 1136 . . 3 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fClus 𝐿))
2 eqid 2735 . . . . . 6 𝐽 = 𝐽
32fclsfil 24034 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐿) → 𝐿 ∈ (Fil‘ 𝐽))
433ad2ant2 1133 . . . 4 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘ 𝐽))
5 fclsfnflim 24051 . . . 4 (𝐿 ∈ (Fil‘ 𝐽) → (𝐴 ∈ (𝐽 fClus 𝐿) ↔ ∃𝑓 ∈ (Fil‘ 𝐽)(𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓))))
64, 5syl 17 . . 3 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fClus 𝐿) ↔ ∃𝑓 ∈ (Fil‘ 𝐽)(𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓))))
71, 6mpbid 232 . 2 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∃𝑓 ∈ (Fil‘ 𝐽)(𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))
8 simpl1 1190 . . . . . 6 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐾 ∈ Top)
9 toptopon2 22940 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
108, 9sylib 218 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐾 ∈ (TopOn‘ 𝐾))
11 simprl 771 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (Fil‘ 𝐽))
12 eqid 2735 . . . . . . . 8 𝐾 = 𝐾
132, 12cnpf 23271 . . . . . . 7 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹: 𝐽 𝐾)
14133ad2ant3 1134 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹: 𝐽 𝐾)
1514adantr 480 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐹: 𝐽 𝐾)
16 flfssfcf 24062 . . . . 5 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑓 ∈ (Fil‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → ((𝐾 fLimf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝑓)‘𝐹))
1710, 11, 15, 16syl3anc 1370 . . . 4 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → ((𝐾 fLimf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝑓)‘𝐹))
1812topopn 22928 . . . . . . . 8 (𝐾 ∈ Top → 𝐾𝐾)
198, 18syl 17 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐾𝐾)
204adantr 480 . . . . . . . 8 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐿 ∈ (Fil‘ 𝐽))
21 filfbas 23872 . . . . . . . 8 (𝐿 ∈ (Fil‘ 𝐽) → 𝐿 ∈ (fBas‘ 𝐽))
2220, 21syl 17 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐿 ∈ (fBas‘ 𝐽))
23 fmfil 23968 . . . . . . 7 (( 𝐾𝐾𝐿 ∈ (fBas‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → (( 𝐾 FilMap 𝐹)‘𝐿) ∈ (Fil‘ 𝐾))
2419, 22, 15, 23syl3anc 1370 . . . . . 6 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → (( 𝐾 FilMap 𝐹)‘𝐿) ∈ (Fil‘ 𝐾))
25 filfbas 23872 . . . . . . . 8 (𝑓 ∈ (Fil‘ 𝐽) → 𝑓 ∈ (fBas‘ 𝐽))
2625ad2antrl 728 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (fBas‘ 𝐽))
27 simprrl 781 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐿𝑓)
28 fmss 23970 . . . . . . 7 ((( 𝐾𝐾𝐿 ∈ (fBas‘ 𝐽) ∧ 𝑓 ∈ (fBas‘ 𝐽)) ∧ (𝐹: 𝐽 𝐾𝐿𝑓)) → (( 𝐾 FilMap 𝐹)‘𝐿) ⊆ (( 𝐾 FilMap 𝐹)‘𝑓))
2919, 22, 26, 15, 27, 28syl32anc 1377 . . . . . 6 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → (( 𝐾 FilMap 𝐹)‘𝐿) ⊆ (( 𝐾 FilMap 𝐹)‘𝑓))
30 fclsss2 24047 . . . . . 6 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ (( 𝐾 FilMap 𝐹)‘𝐿) ∈ (Fil‘ 𝐾) ∧ (( 𝐾 FilMap 𝐹)‘𝐿) ⊆ (( 𝐾 FilMap 𝐹)‘𝑓)) → (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝑓)) ⊆ (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝐿)))
3110, 24, 29, 30syl3anc 1370 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝑓)) ⊆ (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝐿)))
32 fcfval 24057 . . . . . 6 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑓 ∈ (Fil‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝑓)))
3310, 11, 15, 32syl3anc 1370 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝑓)))
34 fcfval 24057 . . . . . 6 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐿 ∈ (Fil‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → ((𝐾 fClusf 𝐿)‘𝐹) = (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝐿)))
3510, 20, 15, 34syl3anc 1370 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → ((𝐾 fClusf 𝐿)‘𝐹) = (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝐿)))
3631, 33, 353sstr4d 4043 . . . 4 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → ((𝐾 fClusf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝐿)‘𝐹))
3717, 36sstrd 4006 . . 3 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → ((𝐾 fLimf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝐿)‘𝐹))
38 simprrr 782 . . . 4 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐴 ∈ (𝐽 fLim 𝑓))
39 simpl3 1192 . . . 4 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
40 cnpflfi 24023 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))
4138, 39, 40syl2anc 584 . . 3 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))
4237, 41sseldd 3996 . 2 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝐿)‘𝐹))
437, 42rexlimddv 3159 1 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝐿)‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  wss 3963   cuni 4912  wf 6559  cfv 6563  (class class class)co 7431  fBascfbas 21370  Topctop 22915  TopOnctopon 22932   CnP ccnp 23249  Filcfil 23869   FilMap cfm 23957   fLim cflim 23958   fLimf cflf 23959   fClus cfcls 23960   fClusf cfcf 23961
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-1o 8505  df-2o 8506  df-map 8867  df-en 8985  df-fin 8988  df-fi 9449  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-cnp 23252  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-fcls 23965  df-fcf 23966
This theorem is referenced by:  cnpfcf  24065
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