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Theorem cnpfcfi 23894
Description: Lemma for cnpfcf 23895. 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)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝐿)β€˜πΉ))
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)β€˜π΄)) β†’ 𝐴 ∈ (𝐽 fClus 𝐿))
2 eqid 2726 . . . . . 6 βˆͺ 𝐽 = βˆͺ 𝐽
32fclsfil 23864 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐿) β†’ 𝐿 ∈ (Filβ€˜βˆͺ 𝐽))
433ad2ant2 1131 . . . 4 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐿 ∈ (Filβ€˜βˆͺ 𝐽))
5 fclsfnflim 23881 . . . 4 (𝐿 ∈ (Filβ€˜βˆͺ 𝐽) β†’ (𝐴 ∈ (𝐽 fClus 𝐿) ↔ βˆƒπ‘“ ∈ (Filβ€˜βˆͺ 𝐽)(𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓))))
64, 5syl 17 . . 3 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (𝐴 ∈ (𝐽 fClus 𝐿) ↔ βˆƒπ‘“ ∈ (Filβ€˜βˆͺ 𝐽)(𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓))))
71, 6mpbid 231 . 2 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ βˆƒπ‘“ ∈ (Filβ€˜βˆͺ 𝐽)(𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))
8 simpl1 1188 . . . . . 6 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ 𝐾 ∈ Top)
9 toptopon2 22770 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
108, 9sylib 217 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾))
11 simprl 768 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ 𝑓 ∈ (Filβ€˜βˆͺ 𝐽))
12 eqid 2726 . . . . . . . 8 βˆͺ 𝐾 = βˆͺ 𝐾
132, 12cnpf 23101 . . . . . . 7 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
14133ad2ant3 1132 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
1514adantr 480 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
16 flfssfcf 23892 . . . . 5 ((𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ 𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾) β†’ ((𝐾 fLimf 𝑓)β€˜πΉ) βŠ† ((𝐾 fClusf 𝑓)β€˜πΉ))
1710, 11, 15, 16syl3anc 1368 . . . 4 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ ((𝐾 fLimf 𝑓)β€˜πΉ) βŠ† ((𝐾 fClusf 𝑓)β€˜πΉ))
1812topopn 22758 . . . . . . . 8 (𝐾 ∈ Top β†’ βˆͺ 𝐾 ∈ 𝐾)
198, 18syl 17 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ βˆͺ 𝐾 ∈ 𝐾)
204adantr 480 . . . . . . . 8 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ 𝐿 ∈ (Filβ€˜βˆͺ 𝐽))
21 filfbas 23702 . . . . . . . 8 (𝐿 ∈ (Filβ€˜βˆͺ 𝐽) β†’ 𝐿 ∈ (fBasβ€˜βˆͺ 𝐽))
2220, 21syl 17 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ 𝐿 ∈ (fBasβ€˜βˆͺ 𝐽))
23 fmfil 23798 . . . . . . 7 ((βˆͺ 𝐾 ∈ 𝐾 ∧ 𝐿 ∈ (fBasβ€˜βˆͺ 𝐽) ∧ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾) β†’ ((βˆͺ 𝐾 FilMap 𝐹)β€˜πΏ) ∈ (Filβ€˜βˆͺ 𝐾))
2419, 22, 15, 23syl3anc 1368 . . . . . 6 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ ((βˆͺ 𝐾 FilMap 𝐹)β€˜πΏ) ∈ (Filβ€˜βˆͺ 𝐾))
25 filfbas 23702 . . . . . . . 8 (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) β†’ 𝑓 ∈ (fBasβ€˜βˆͺ 𝐽))
2625ad2antrl 725 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ 𝑓 ∈ (fBasβ€˜βˆͺ 𝐽))
27 simprrl 778 . . . . . . 7 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ 𝐿 βŠ† 𝑓)
28 fmss 23800 . . . . . . 7 (((βˆͺ 𝐾 ∈ 𝐾 ∧ 𝐿 ∈ (fBasβ€˜βˆͺ 𝐽) ∧ 𝑓 ∈ (fBasβ€˜βˆͺ 𝐽)) ∧ (𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾 ∧ 𝐿 βŠ† 𝑓)) β†’ ((βˆͺ 𝐾 FilMap 𝐹)β€˜πΏ) βŠ† ((βˆͺ 𝐾 FilMap 𝐹)β€˜π‘“))
2919, 22, 26, 15, 27, 28syl32anc 1375 . . . . . 6 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ ((βˆͺ 𝐾 FilMap 𝐹)β€˜πΏ) βŠ† ((βˆͺ 𝐾 FilMap 𝐹)β€˜π‘“))
30 fclsss2 23877 . . . . . 6 ((𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ ((βˆͺ 𝐾 FilMap 𝐹)β€˜πΏ) ∈ (Filβ€˜βˆͺ 𝐾) ∧ ((βˆͺ 𝐾 FilMap 𝐹)β€˜πΏ) βŠ† ((βˆͺ 𝐾 FilMap 𝐹)β€˜π‘“)) β†’ (𝐾 fClus ((βˆͺ 𝐾 FilMap 𝐹)β€˜π‘“)) βŠ† (𝐾 fClus ((βˆͺ 𝐾 FilMap 𝐹)β€˜πΏ)))
3110, 24, 29, 30syl3anc 1368 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ (𝐾 fClus ((βˆͺ 𝐾 FilMap 𝐹)β€˜π‘“)) βŠ† (𝐾 fClus ((βˆͺ 𝐾 FilMap 𝐹)β€˜πΏ)))
32 fcfval 23887 . . . . . 6 ((𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ 𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾) β†’ ((𝐾 fClusf 𝑓)β€˜πΉ) = (𝐾 fClus ((βˆͺ 𝐾 FilMap 𝐹)β€˜π‘“)))
3310, 11, 15, 32syl3anc 1368 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ ((𝐾 fClusf 𝑓)β€˜πΉ) = (𝐾 fClus ((βˆͺ 𝐾 FilMap 𝐹)β€˜π‘“)))
34 fcfval 23887 . . . . . 6 ((𝐾 ∈ (TopOnβ€˜βˆͺ 𝐾) ∧ 𝐿 ∈ (Filβ€˜βˆͺ 𝐽) ∧ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾) β†’ ((𝐾 fClusf 𝐿)β€˜πΉ) = (𝐾 fClus ((βˆͺ 𝐾 FilMap 𝐹)β€˜πΏ)))
3510, 20, 15, 34syl3anc 1368 . . . . 5 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ ((𝐾 fClusf 𝐿)β€˜πΉ) = (𝐾 fClus ((βˆͺ 𝐾 FilMap 𝐹)β€˜πΏ)))
3631, 33, 353sstr4d 4024 . . . 4 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ ((𝐾 fClusf 𝑓)β€˜πΉ) βŠ† ((𝐾 fClusf 𝐿)β€˜πΉ))
3717, 36sstrd 3987 . . 3 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ ((𝐾 fLimf 𝑓)β€˜πΉ) βŠ† ((𝐾 fClusf 𝐿)β€˜πΉ))
38 simprrr 779 . . . 4 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ 𝐴 ∈ (𝐽 fLim 𝑓))
39 simpl3 1190 . . . 4 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄))
40 cnpflfi 23853 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))
4138, 39, 40syl2anc 583 . . 3 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))
4237, 41sseldd 3978 . 2 (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) ∧ (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) ∧ (𝐿 βŠ† 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝐿)β€˜πΉ))
437, 42rexlimddv 3155 1 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fClusf 𝐿)β€˜πΉ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064   βŠ† wss 3943  βˆͺ cuni 4902  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404  fBascfbas 21223  Topctop 22745  TopOnctopon 22762   CnP ccnp 23079  Filcfil 23699   FilMap cfm 23787   fLim cflim 23788   fLimf cflf 23789   fClus cfcls 23790   fClusf cfcf 23791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-1o 8464  df-er 8702  df-map 8821  df-en 8939  df-fin 8942  df-fi 9405  df-fbas 21232  df-fg 21233  df-top 22746  df-topon 22763  df-cld 22873  df-ntr 22874  df-cls 22875  df-nei 22952  df-cnp 23082  df-fil 23700  df-fm 23792  df-flim 23793  df-flf 23794  df-fcls 23795  df-fcf 23796
This theorem is referenced by:  cnpfcf  23895
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