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Theorem cnpflf 23725
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 22974 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
213expa 1118 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
323adantl3 1168 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
4 cnpflfi 23723 . . . . . . 7 ((𝐴 ∈ (𝐽 fLim 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄)) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))
54expcom 414 . . . . . 6 (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π΄) β†’ (𝐴 ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π΄) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)))
65ralrimivw 3150 . . . . 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 482 . . . 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 512 . . 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 413 . 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 1191 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
11 simpl3 1193 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐴 ∈ 𝑋)
12 neiflim 23698 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ (𝐽 fLim ((neiβ€˜π½)β€˜{𝐴})))
1310, 11, 12syl2anc 584 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝐴 ∈ (𝐽 fLim ((neiβ€˜π½)β€˜{𝐴})))
1411snssd 4812 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ {𝐴} βŠ† 𝑋)
1511snn0d 4779 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ {𝐴} β‰  βˆ…)
16 neifil 23604 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ {𝐴} βŠ† 𝑋 ∧ {𝐴} β‰  βˆ…) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
1710, 14, 15, 16syl3anc 1371 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((neiβ€˜π½)β€˜{𝐴}) ∈ (Filβ€˜π‘‹))
18 oveq2 7419 . . . . . . . . 9 (𝑓 = ((neiβ€˜π½)β€˜{𝐴}) β†’ (𝐽 fLim 𝑓) = (𝐽 fLim ((neiβ€˜π½)β€˜{𝐴})))
1918eleq2d 2819 . . . . . . . 8 (𝑓 = ((neiβ€˜π½)β€˜{𝐴}) β†’ (𝐴 ∈ (𝐽 fLim 𝑓) ↔ 𝐴 ∈ (𝐽 fLim ((neiβ€˜π½)β€˜{𝐴}))))
20 oveq2 7419 . . . . . . . . . 10 (𝑓 = ((neiβ€˜π½)β€˜{𝐴}) β†’ (𝐾 fLimf 𝑓) = (𝐾 fLimf ((neiβ€˜π½)β€˜{𝐴})))
2120fveq1d 6893 . . . . . . . . 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 2819 . . . . . . . 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 344 . . . . . . 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 3608 . . . . . 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 572 . . 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 2732 . . . 4 ((neiβ€˜π½)β€˜{𝐴}) = ((neiβ€˜π½)β€˜{𝐴})
2928cnpflf2 23724 . . 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 258 . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  TopOnctopon 22632  neicnei 22821   CnP ccnp 22949  Filcfil 23569   fLim cflim 23658   fLimf cflf 23659
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-fbas 21141  df-fg 21142  df-top 22616  df-topon 22633  df-ntr 22744  df-nei 22822  df-cnp 22952  df-fil 23570  df-fm 23662  df-flim 23663  df-flf 23664
This theorem is referenced by:  cnflf  23726  cnpfcf  23765
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