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Theorem cnpfcfi 21654
Description: Lemma for cnpfcf 21655. 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 1055 . . 3 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fClus 𝐿))
2 eqid 2610 . . . . . 6 𝐽 = 𝐽
32fclsfil 21624 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐿) → 𝐿 ∈ (Fil‘ 𝐽))
433ad2ant2 1076 . . . 4 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘ 𝐽))
5 fclsfnflim 21641 . . . 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 221 . 2 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∃𝑓 ∈ (Fil‘ 𝐽)(𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))
8 simpl1 1057 . . . . . 6 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐾 ∈ Top)
9 eqid 2610 . . . . . . 7 𝐾 = 𝐾
109toptopon 20548 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
118, 10sylib 207 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐾 ∈ (TopOn‘ 𝐾))
12 simprl 790 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (Fil‘ 𝐽))
132, 9cnpf 20861 . . . . . . 7 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹: 𝐽 𝐾)
14133ad2ant3 1077 . . . . . 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 21652 . . . . 5 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑓 ∈ (Fil‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → ((𝐾 fLimf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝑓)‘𝐹))
1711, 12, 15, 16syl3anc 1318 . . . 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))
189topopn 20536 . . . . . . . 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 21462 . . . . . . . 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 21558 . . . . . . 7 (( 𝐾𝐾𝐿 ∈ (fBas‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → (( 𝐾 FilMap 𝐹)‘𝐿) ∈ (Fil‘ 𝐾))
2419, 22, 15, 23syl3anc 1318 . . . . . 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 21462 . . . . . . . 8 (𝑓 ∈ (Fil‘ 𝐽) → 𝑓 ∈ (fBas‘ 𝐽))
2625ad2antrl 760 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (fBas‘ 𝐽))
27 simprrl 800 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐿𝑓)
28 fmss 21560 . . . . . . 7 ((( 𝐾𝐾𝐿 ∈ (fBas‘ 𝐽) ∧ 𝑓 ∈ (fBas‘ 𝐽)) ∧ (𝐹: 𝐽 𝐾𝐿𝑓)) → (( 𝐾 FilMap 𝐹)‘𝐿) ⊆ (( 𝐾 FilMap 𝐹)‘𝑓))
2919, 22, 26, 15, 27, 28syl32anc 1326 . . . . . 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 21637 . . . . . 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)))
3111, 24, 29, 30syl3anc 1318 . . . . 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 21647 . . . . . 6 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑓 ∈ (Fil‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝑓)))
3311, 12, 15, 32syl3anc 1318 . . . . 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 21647 . . . . . 6 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐿 ∈ (Fil‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → ((𝐾 fClusf 𝐿)‘𝐹) = (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝐿)))
3511, 20, 15, 34syl3anc 1318 . . . . 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 3611 . . . 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 3578 . . 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 801 . . . 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 1059 . . . 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 21613 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))
4138, 39, 40syl2anc 691 . . 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 3569 . 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 3017 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wrex 2897  wss 3540   cuni 4372  wf 5800  cfv 5804  (class class class)co 6549  fBascfbas 19555  Topctop 20517  TopOnctopon 20518   CnP ccnp 20839  Filcfil 21459   FilMap cfm 21547   fLim cflim 21548   fLimf cflf 21549   fClus cfcls 21550   fClusf cfcf 21551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-fin 7845  df-fi 8200  df-fbas 19564  df-fg 19565  df-top 20521  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-cnp 20842  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-fcls 21555  df-fcf 21556
This theorem is referenced by:  cnpfcf  21655
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