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Theorem cnpfcfi 23535
Description: Lemma for cnpfcf 23536. 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 1137 . . 3 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐴 ∈ (𝐽 fClus 𝐿))
2 eqid 2732 . . . . . 6 βˆͺ 𝐽 = βˆͺ 𝐽
32fclsfil 23505 . . . . 5 (𝐴 ∈ (𝐽 fClus 𝐿) β†’ 𝐿 ∈ (Filβ€˜βˆͺ 𝐽))
433ad2ant2 1134 . . . 4 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐿 ∈ (Filβ€˜βˆͺ 𝐽))
5 fclsfnflim 23522 . . . 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 1191 . . . . . 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 22411 . . . . . 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 2732 . . . . . . . 8 βˆͺ 𝐾 = βˆͺ 𝐾
132, 12cnpf 22742 . . . . . . 7 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
14133ad2ant3 1135 . . . . . 6 ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
1514adantr 481 . . . . 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 23533 . . . . 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 1371 . . . 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 22399 . . . . . . . 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 481 . . . . . . . 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 23343 . . . . . . . 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 23439 . . . . . . 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 1371 . . . . . 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 23343 . . . . . . . 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 23441 . . . . . . 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 1378 . . . . . 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 23518 . . . . . 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 1371 . . . . 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 23528 . . . . . 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 1371 . . . . 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 23528 . . . . . 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 1371 . . . . 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 4028 . . . 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 3991 . . 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 1193 . . . 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 23494 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))
4138, 39, 40syl2anc 584 . . 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 3982 . 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 3161 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   βŠ† wss 3947  βˆͺ cuni 4907  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  fBascfbas 20924  Topctop 22386  TopOnctopon 22403   CnP ccnp 22720  Filcfil 23340   FilMap cfm 23428   fLim cflim 23429   fLimf cflf 23430   fClus cfcls 23431   fClusf cfcf 23432
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-1o 8462  df-er 8699  df-map 8818  df-en 8936  df-fin 8939  df-fi 9402  df-fbas 20933  df-fg 20934  df-top 22387  df-topon 22404  df-cld 22514  df-ntr 22515  df-cls 22516  df-nei 22593  df-cnp 22723  df-fil 23341  df-fm 23433  df-flim 23434  df-flf 23435  df-fcls 23436  df-fcf 23437
This theorem is referenced by:  cnpfcf  23536
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