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Theorem cnfcf 23416
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 22654 . 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 23415 . . . . . . 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 1174 . . . . . 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 3169 . . . 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 3271 . . . . 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 2733 . . . . . . . . . . . 12 βˆͺ 𝐽 = βˆͺ 𝐽
98fclselbas 23390 . . . . . . . . . . 11 (π‘₯ ∈ (𝐽 fClus 𝑓) β†’ π‘₯ ∈ βˆͺ 𝐽)
10 toponuni 22286 . . . . . . . . . . . . 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 2820 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ βˆͺ 𝐽))
139, 12syl5ibr 246 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (π‘₯ ∈ (𝐽 fClus 𝑓) β†’ π‘₯ ∈ 𝑋))
1413pm4.71rd 564 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (π‘₯ ∈ (𝐽 fClus 𝑓) ↔ (π‘₯ ∈ 𝑋 ∧ π‘₯ ∈ (𝐽 fClus 𝑓))))
1514imbi1d 342 . . . . . . . 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 452 . . . . . . . 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 3167 . . . . . 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 3171 . . . . 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 580 . 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆͺ cuni 4869  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  TopOnctopon 22282   Cn ccn 22598   CnP ccnp 22599  Filcfil 23219   fClus cfcls 23310   fClusf cfcf 23311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-fin 8893  df-fi 9355  df-topgen 17333  df-fbas 20816  df-fg 20817  df-top 22266  df-topon 22283  df-cld 22393  df-ntr 22394  df-cls 22395  df-nei 22472  df-cn 22601  df-cnp 22602  df-fil 23220  df-fm 23312  df-flim 23313  df-flf 23314  df-fcls 23315  df-fcf 23316
This theorem is referenced by: (None)
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