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Theorem cnfcf 23939
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 23177 . 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 768 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
3 cnpfcf 23938 . . . . . . 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 1171 . . . . . 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 706 . . . . 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 3170 . . . 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 3281 . . . . 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 2727 . . . . . . . . . . . 12 βˆͺ 𝐽 = βˆͺ 𝐽
98fclselbas 23913 . . . . . . . . . . 11 (π‘₯ ∈ (𝐽 fClus 𝑓) β†’ π‘₯ ∈ βˆͺ 𝐽)
10 toponuni 22809 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
1110ad2antrr 725 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝑋 = βˆͺ 𝐽)
1211eleq2d 2814 . . . . . . . . . . 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 3168 . . . . . 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 3172 . . . . 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 1534   ∈ wcel 2099  βˆ€wral 3056  βˆͺ cuni 4903  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414  TopOnctopon 22805   Cn ccn 23121   CnP ccnp 23122  Filcfil 23742   fClus cfcls 23833   fClusf cfcf 23834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-1o 8480  df-er 8718  df-map 8840  df-en 8958  df-fin 8961  df-fi 9428  df-topgen 17418  df-fbas 21269  df-fg 21270  df-top 22789  df-topon 22806  df-cld 22916  df-ntr 22917  df-cls 22918  df-nei 22995  df-cn 23124  df-cnp 23125  df-fil 23743  df-fm 23835  df-flim 23836  df-flf 23837  df-fcls 23838  df-fcf 23839
This theorem is referenced by: (None)
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