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Theorem cnpflf 23505
Description: Continuity of a function at a point in terms of filter limits. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
Assertion
Ref Expression
cnpflf ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)))))
Distinct variable groups:   𝐴,𝑓   𝑓,𝑋   𝑓,π‘Œ   𝑓,𝐹   𝑓,𝐽   𝑓,𝐾

Proof of Theorem cnpflf
StepHypRef Expression
1 cnpf2 22754 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
213expa 1119 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
323adantl3 1169 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
4 cnpflfi 23503 . . . . . . 7 ((𝐴 ∈ (𝐽 fLim 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))
54expcom 415 . . . . . 6 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) β†’ (𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)))
65ralrimivw 3151 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) β†’ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)))
76adantl 483 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)))
83, 7jca 513 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))))
98ex 414 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)))))
10 simpl1 1192 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
11 simpl3 1194 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐴 ∈ 𝑋)
12 neiflim 23478 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ (𝐽 fLim ((neiβ€˜π½)β€˜{𝐴})))
1310, 11, 12syl2anc 585 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐴 ∈ (𝐽 fLim ((neiβ€˜π½)β€˜{𝐴})))
1411snssd 4813 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ {𝐴} βŠ† 𝑋)
1511snn0d 4780 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ {𝐴} β‰  βˆ…)
16 neifil 23384 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ {𝐴} βŠ† 𝑋 ∧ {𝐴} β‰  βˆ…) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
1710, 14, 15, 16syl3anc 1372 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
18 oveq2 7417 . . . . . . . . 9 (𝑓 = ((neiβ€˜π½)β€˜{𝐴}) β†’ (𝐽 fLim 𝑓) = (𝐽 fLim ((neiβ€˜π½)β€˜{𝐴})))
1918eleq2d 2820 . . . . . . . 8 (𝑓 = ((neiβ€˜π½)β€˜{𝐴}) β†’ (𝐴 ∈ (𝐽 fLim 𝑓) ↔ 𝐴 ∈ (𝐽 fLim ((neiβ€˜π½)β€˜{𝐴}))))
20 oveq2 7417 . . . . . . . . . 10 (𝑓 = ((neiβ€˜π½)β€˜{𝐴}) β†’ (𝐾 fLimf 𝑓) = (𝐾 fLimf ((neiβ€˜π½)β€˜{𝐴})))
2120fveq1d 6894 . . . . . . . . 9 (𝑓 = ((neiβ€˜π½)β€˜{𝐴}) β†’ ((𝐾 fLimf 𝑓)β€˜πΉ) = ((𝐾 fLimf ((neiβ€˜π½)β€˜{𝐴}))β€˜πΉ))
2221eleq2d 2820 . . . . . . . 8 (𝑓 = ((neiβ€˜π½)β€˜{𝐴}) β†’ ((πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ) ↔ (πΉβ€˜π΄) ∈ ((𝐾 fLimf ((neiβ€˜π½)β€˜{𝐴}))β€˜πΉ)))
2319, 22imbi12d 345 . . . . . . 7 (𝑓 = ((neiβ€˜π½)β€˜{𝐴}) β†’ ((𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)) ↔ (𝐴 ∈ (𝐽 fLim ((neiβ€˜π½)β€˜{𝐴})) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf ((neiβ€˜π½)β€˜{𝐴}))β€˜πΉ))))
2423rspcv 3609 . . . . . 6 (((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)) β†’ (𝐴 ∈ (𝐽 fLim ((neiβ€˜π½)β€˜{𝐴})) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf ((neiβ€˜π½)β€˜{𝐴}))β€˜πΉ))))
2517, 24syl 17 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)) β†’ (𝐴 ∈ (𝐽 fLim ((neiβ€˜π½)β€˜{𝐴})) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf ((neiβ€˜π½)β€˜{𝐴}))β€˜πΉ))))
2613, 25mpid 44 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf ((neiβ€˜π½)β€˜{𝐴}))β€˜πΉ)))
2726imdistanda 573 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ (πΉβ€˜π΄) ∈ ((𝐾 fLimf ((neiβ€˜π½)β€˜{𝐴}))β€˜πΉ))))
28 eqid 2733 . . . 4 ((neiβ€˜π½)β€˜{𝐴}) = ((neiβ€˜π½)β€˜{𝐴})
2928cnpflf2 23504 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ (πΉβ€˜π΄) ∈ ((𝐾 fLimf ((neiβ€˜π½)β€˜{𝐴}))β€˜πΉ))))
3027, 29sylibrd 259 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))) β†’ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)))
319, 30impbid 211 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)(𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  TopOnctopon 22412  neicnei 22601   CnP ccnp 22729  Filcfil 23349   fLim cflim 23438   fLimf cflf 23439
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-fbas 20941  df-fg 20942  df-top 22396  df-topon 22413  df-ntr 22524  df-nei 22602  df-cnp 22732  df-fil 23350  df-fm 23442  df-flim 23443  df-flf 23444
This theorem is referenced by:  cnflf  23506  cnpfcf  23545
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