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Theorem cnpfcfi 23964
Description: Lemma for cnpfcf 23965. 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 2728 . . . . . 6 βˆͺ 𝐽 = βˆͺ 𝐽
32fclsfil 23934 . . . . 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 23951 . . . 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 22840 . . . . . 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 769 . . . . 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 2728 . . . . . . . 8 βˆͺ 𝐾 = βˆͺ 𝐾
132, 12cnpf 23171 . . . . . . 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 479 . . . . 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 23962 . . . . 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 22828 . . . . . . . 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 479 . . . . . . . 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 23772 . . . . . . . 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 23868 . . . . . . 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 23772 . . . . . . . 8 (𝑓 ∈ (Filβ€˜βˆͺ 𝐽) β†’ 𝑓 ∈ (fBasβ€˜βˆͺ 𝐽))
2625ad2antrl 726 . . . . . . 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 779 . . . . . . 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 23870 . . . . . . 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 23947 . . . . . 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 23957 . . . . . 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 23957 . . . . . 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 4029 . . . 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 3992 . . 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 780 . . . 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 23923 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))
4138, 39, 40syl2anc 582 . . 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 3983 . 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 3158 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3067   βŠ† wss 3949  βˆͺ cuni 4912  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  fBascfbas 21274  Topctop 22815  TopOnctopon 22832   CnP ccnp 23149  Filcfil 23769   FilMap cfm 23857   fLim cflim 23858   fLimf cflf 23859   fClus cfcls 23860   fClusf cfcf 23861
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-1o 8493  df-er 8731  df-map 8853  df-en 8971  df-fin 8974  df-fi 9442  df-fbas 21283  df-fg 21284  df-top 22816  df-topon 22833  df-cld 22943  df-ntr 22944  df-cls 22945  df-nei 23022  df-cnp 23152  df-fil 23770  df-fm 23862  df-flim 23863  df-flf 23864  df-fcls 23865  df-fcf 23866
This theorem is referenced by:  cnpfcf  23965
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