MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnpfcfi Structured version   Visualization version   GIF version

Theorem cnpfcfi 22576
Description: Lemma for cnpfcf 22577. 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 1129 . . 3 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fClus 𝐿))
2 eqid 2818 . . . . . 6 𝐽 = 𝐽
32fclsfil 22546 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐿) → 𝐿 ∈ (Fil‘ 𝐽))
433ad2ant2 1126 . . . 4 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘ 𝐽))
5 fclsfnflim 22563 . . . 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 233 . 2 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∃𝑓 ∈ (Fil‘ 𝐽)(𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))
8 simpl1 1183 . . . . . 6 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐾 ∈ Top)
9 toptopon2 21454 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
108, 9sylib 219 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐾 ∈ (TopOn‘ 𝐾))
11 simprl 767 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (Fil‘ 𝐽))
12 eqid 2818 . . . . . . . 8 𝐾 = 𝐾
132, 12cnpf 21783 . . . . . . 7 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹: 𝐽 𝐾)
14133ad2ant3 1127 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹: 𝐽 𝐾)
1514adantr 481 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐹: 𝐽 𝐾)
16 flfssfcf 22574 . . . . 5 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑓 ∈ (Fil‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → ((𝐾 fLimf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝑓)‘𝐹))
1710, 11, 15, 16syl3anc 1363 . . . 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 21442 . . . . . . . 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 481 . . . . . . . 8 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐿 ∈ (Fil‘ 𝐽))
21 filfbas 22384 . . . . . . . 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 22480 . . . . . . 7 (( 𝐾𝐾𝐿 ∈ (fBas‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → (( 𝐾 FilMap 𝐹)‘𝐿) ∈ (Fil‘ 𝐾))
2419, 22, 15, 23syl3anc 1363 . . . . . 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 22384 . . . . . . . 8 (𝑓 ∈ (Fil‘ 𝐽) → 𝑓 ∈ (fBas‘ 𝐽))
2625ad2antrl 724 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (fBas‘ 𝐽))
27 simprrl 777 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐿𝑓)
28 fmss 22482 . . . . . . 7 ((( 𝐾𝐾𝐿 ∈ (fBas‘ 𝐽) ∧ 𝑓 ∈ (fBas‘ 𝐽)) ∧ (𝐹: 𝐽 𝐾𝐿𝑓)) → (( 𝐾 FilMap 𝐹)‘𝐿) ⊆ (( 𝐾 FilMap 𝐹)‘𝑓))
2919, 22, 26, 15, 27, 28syl32anc 1370 . . . . . 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 22559 . . . . . 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 1363 . . . . 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 22569 . . . . . 6 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑓 ∈ (Fil‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝑓)))
3310, 11, 15, 32syl3anc 1363 . . . . 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 22569 . . . . . 6 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐿 ∈ (Fil‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → ((𝐾 fClusf 𝐿)‘𝐹) = (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝐿)))
3510, 20, 15, 34syl3anc 1363 . . . . 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 4011 . . . 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 3974 . . 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 778 . . . 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 1185 . . . 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 22535 . . . 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 3965 . 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 3288 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  wss 3933   cuni 4830  wf 6344  cfv 6348  (class class class)co 7145  fBascfbas 20461  Topctop 21429  TopOnctopon 21446   CnP ccnp 21761  Filcfil 22381   FilMap cfm 22469   fLim cflim 22470   fLimf cflf 22471   fClus cfcls 22472   fClusf cfcf 22473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-fin 8501  df-fi 8863  df-fbas 20470  df-fg 20471  df-top 21430  df-topon 21447  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-cnp 21764  df-fil 22382  df-fm 22474  df-flim 22475  df-flf 22476  df-fcls 22477  df-fcf 22478
This theorem is referenced by:  cnpfcf  22577
  Copyright terms: Public domain W3C validator