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Theorem cnfcf 23868
Description: Continuity of a function in terms of cluster points of a function. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
cnfcf ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ (𝐽 fClus 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))))
Distinct variable groups:   π‘₯,𝑓,𝐹   𝑓,𝐽,π‘₯   𝑓,𝐾,π‘₯   𝑓,𝑋,π‘₯   𝑓,π‘Œ,π‘₯
Proof of Theorem cnfcf
StepHypRef Expression
1 cncnp 23106 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘₯))))
2 simplr 766 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
3 cnpfcf 23867 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘₯) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(π‘₯ ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)))))
43ad4ant124 1170 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘₯) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(π‘₯ ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)))))
52, 4mpbirand 704 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘₯) ↔ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(π‘₯ ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))))
65ralbidva 3167 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘₯ ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘₯) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘“ ∈ (Filβ€˜π‘‹)(π‘₯ ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))))
7 ralcom 3278 . . . . 5 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘“ ∈ (Filβ€˜π‘‹)(π‘₯ ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) ↔ βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ 𝑋 (π‘₯ ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)))
8 eqid 2724 . . . . . . . . . . . 12 βˆͺ 𝐽 = βˆͺ 𝐽
98fclselbas 23842 . . . . . . . . . . 11 (π‘₯ ∈ (𝐽 fClus 𝑓) β†’ π‘₯ ∈ βˆͺ 𝐽)
10 toponuni 22738 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
1110ad2antrr 723 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝑋 = βˆͺ 𝐽)
1211eleq2d 2811 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ βˆͺ 𝐽))
139, 12imbitrrid 245 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (π‘₯ ∈ (𝐽 fClus 𝑓) β†’ π‘₯ ∈ 𝑋))
1413pm4.71rd 562 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (π‘₯ ∈ (𝐽 fClus 𝑓) ↔ (π‘₯ ∈ 𝑋 ∧ π‘₯ ∈ (𝐽 fClus 𝑓))))
1514imbi1d 341 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((π‘₯ ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) ↔ ((π‘₯ ∈ 𝑋 ∧ π‘₯ ∈ (𝐽 fClus 𝑓)) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))))
16 impexp 450 . . . . . . . 8 (((π‘₯ ∈ 𝑋 ∧ π‘₯ ∈ (𝐽 fClus 𝑓)) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) ↔ (π‘₯ ∈ 𝑋 β†’ (π‘₯ ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))))
1715, 16bitrdi 287 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((π‘₯ ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) ↔ (π‘₯ ∈ 𝑋 β†’ (π‘₯ ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)))))
1817ralbidv2 3165 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘₯ ∈ (𝐽 fClus 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ) ↔ βˆ€π‘₯ ∈ 𝑋 (π‘₯ ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))))
1918ralbidv 3169 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ (𝐽 fClus 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ) ↔ βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ 𝑋 (π‘₯ ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))))
207, 19bitr4id 290 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘“ ∈ (Filβ€˜π‘‹)(π‘₯ ∈ (𝐽 fClus 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)) ↔ βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ (𝐽 fClus 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)))
216, 20bitrd 279 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘₯ ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘₯) ↔ βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ (𝐽 fClus 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ)))
2221pm5.32da 578 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘₯)) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ (𝐽 fClus 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))))
231, 22bitrd 279 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ (𝐽 fClus 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fClusf 𝑓)β€˜πΉ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βˆͺ cuni 4899  βŸΆwf 6529  β€˜cfv 6533  (class class class)co 7401  TopOnctopon 22734   Cn ccn 23050   CnP ccnp 23051  Filcfil 23671   fClus cfcls 23762   fClusf cfcf 23763
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 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-1o 8461  df-er 8699  df-map 8818  df-en 8936  df-fin 8939  df-fi 9402  df-topgen 17388  df-fbas 21225  df-fg 21226  df-top 22718  df-topon 22735  df-cld 22845  df-ntr 22846  df-cls 22847  df-nei 22924  df-cn 23053  df-cnp 23054  df-fil 23672  df-fm 23764  df-flim 23765  df-flf 23766  df-fcls 23767  df-fcf 23768
This theorem is referenced by: (None)
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