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Theorem cnpfcfi 22624
 Description: Lemma for cnpfcf 22625. 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 1134 . . 3 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fClus 𝐿))
2 eqid 2821 . . . . . 6 𝐽 = 𝐽
32fclsfil 22594 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐿) → 𝐿 ∈ (Fil‘ 𝐽))
433ad2ant2 1131 . . . 4 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘ 𝐽))
5 fclsfnflim 22611 . . . 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 235 . 2 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∃𝑓 ∈ (Fil‘ 𝐽)(𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))
8 simpl1 1188 . . . . . 6 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐾 ∈ Top)
9 toptopon2 21502 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
108, 9sylib 221 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐾 ∈ (TopOn‘ 𝐾))
11 simprl 770 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (Fil‘ 𝐽))
12 eqid 2821 . . . . . . . 8 𝐾 = 𝐾
132, 12cnpf 21831 . . . . . . 7 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹: 𝐽 𝐾)
14133ad2ant3 1132 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹: 𝐽 𝐾)
1514adantr 484 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐹: 𝐽 𝐾)
16 flfssfcf 22622 . . . . 5 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑓 ∈ (Fil‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → ((𝐾 fLimf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝑓)‘𝐹))
1710, 11, 15, 16syl3anc 1368 . . . 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 21490 . . . . . . . 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 484 . . . . . . . 8 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐿 ∈ (Fil‘ 𝐽))
21 filfbas 22432 . . . . . . . 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 22528 . . . . . . 7 (( 𝐾𝐾𝐿 ∈ (fBas‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → (( 𝐾 FilMap 𝐹)‘𝐿) ∈ (Fil‘ 𝐾))
2419, 22, 15, 23syl3anc 1368 . . . . . 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 22432 . . . . . . . 8 (𝑓 ∈ (Fil‘ 𝐽) → 𝑓 ∈ (fBas‘ 𝐽))
2625ad2antrl 727 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (fBas‘ 𝐽))
27 simprrl 780 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘ 𝐽) ∧ (𝐿𝑓𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐿𝑓)
28 fmss 22530 . . . . . . 7 ((( 𝐾𝐾𝐿 ∈ (fBas‘ 𝐽) ∧ 𝑓 ∈ (fBas‘ 𝐽)) ∧ (𝐹: 𝐽 𝐾𝐿𝑓)) → (( 𝐾 FilMap 𝐹)‘𝐿) ⊆ (( 𝐾 FilMap 𝐹)‘𝑓))
2919, 22, 26, 15, 27, 28syl32anc 1375 . . . . . 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 22607 . . . . . 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 1368 . . . . 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 22617 . . . . . 6 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝑓 ∈ (Fil‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝑓)))
3310, 11, 15, 32syl3anc 1368 . . . . 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 22617 . . . . . 6 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐿 ∈ (Fil‘ 𝐽) ∧ 𝐹: 𝐽 𝐾) → ((𝐾 fClusf 𝐿)‘𝐹) = (𝐾 fClus (( 𝐾 FilMap 𝐹)‘𝐿)))
3510, 20, 15, 34syl3anc 1368 . . . . 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 3990 . . . 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 3953 . . 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 781 . . . 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 1190 . . . 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 22583 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))
4138, 39, 40syl2anc 587 . . 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 3944 . 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 3277 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∃wrex 3127   ⊆ wss 3910  ∪ cuni 4811  ⟶wf 6324  ‘cfv 6328  (class class class)co 7130  fBascfbas 20509  Topctop 21477  TopOnctopon 21494   CnP ccnp 21809  Filcfil 22429   FilMap cfm 22517   fLim cflim 22518   fLimf cflf 22519   fClus cfcls 22520   fClusf cfcf 22521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-map 8383  df-en 8485  df-fin 8488  df-fi 8851  df-fbas 20518  df-fg 20519  df-top 21478  df-topon 21495  df-cld 21603  df-ntr 21604  df-cls 21605  df-nei 21682  df-cnp 21812  df-fil 22430  df-fm 22522  df-flim 22523  df-flf 22524  df-fcls 22525  df-fcf 22526 This theorem is referenced by:  cnpfcf  22625
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