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Theorem cnflf 23505
Description: A function is continuous iff it respects filter limits. (Contributed by Jeff Hankins, 6-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
Assertion
Ref Expression
cnflf ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ (𝐽 fLim 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))))
Distinct variable groups:   π‘₯,𝑓,𝑋   𝑓,π‘Œ,π‘₯   𝑓,𝐹,π‘₯   𝑓,𝐽,π‘₯   𝑓,𝐾,π‘₯
Proof of Theorem cnflf
StepHypRef Expression
1 cncnp 22783 . 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 767 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
3 cnpflf 23504 . . . . . . 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 fLim 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)))))
43ad4ant124 1173 . . . . . 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 fLim 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)))))
52, 4mpbirand 705 . . . . 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 fLim 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))))
65ralbidva 3175 . . . 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 fLim 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))))
7 eqid 2732 . . . . . . . . . . . 12 βˆͺ 𝐽 = βˆͺ 𝐽
87flimelbas 23471 . . . . . . . . . . 11 (π‘₯ ∈ (𝐽 fLim 𝑓) β†’ π‘₯ ∈ βˆͺ 𝐽)
9 toponuni 22415 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
109ad2antrr 724 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝑋 = βˆͺ 𝐽)
1110eleq2d 2819 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (π‘₯ ∈ 𝑋 ↔ π‘₯ ∈ βˆͺ 𝐽))
128, 11imbitrrid 245 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (π‘₯ ∈ (𝐽 fLim 𝑓) β†’ π‘₯ ∈ 𝑋))
1312pm4.71rd 563 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (π‘₯ ∈ (𝐽 fLim 𝑓) ↔ (π‘₯ ∈ 𝑋 ∧ π‘₯ ∈ (𝐽 fLim 𝑓))))
1413imbi1d 341 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((π‘₯ ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)) ↔ ((π‘₯ ∈ 𝑋 ∧ π‘₯ ∈ (𝐽 fLim 𝑓)) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))))
15 impexp 451 . . . . . . . 8 (((π‘₯ ∈ 𝑋 ∧ π‘₯ ∈ (𝐽 fLim 𝑓)) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)) ↔ (π‘₯ ∈ 𝑋 β†’ (π‘₯ ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))))
1614, 15bitrdi 286 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((π‘₯ ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)) ↔ (π‘₯ ∈ 𝑋 β†’ (π‘₯ ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)))))
1716ralbidv2 3173 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘₯ ∈ (𝐽 fLim 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ) ↔ βˆ€π‘₯ ∈ 𝑋 (π‘₯ ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))))
1817ralbidv 3177 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ (𝐽 fLim 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ) ↔ βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ 𝑋 (π‘₯ ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))))
19 ralcom 3286 . . . . 5 (βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ 𝑋 (π‘₯ ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘“ ∈ (Filβ€˜π‘‹)(π‘₯ ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)))
2018, 19bitrdi 286 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ (𝐽 fLim 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘“ ∈ (Filβ€˜π‘‹)(π‘₯ ∈ (𝐽 fLim 𝑓) β†’ (πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))))
216, 20bitr4d 281 . . 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 fLim 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ)))
2221pm5.32da 579 . 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 fLim 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))))
231, 22bitrd 278 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘“ ∈ (Filβ€˜π‘‹)βˆ€π‘₯ ∈ (𝐽 fLim 𝑓)(πΉβ€˜π‘₯) ∈ ((𝐾 fLimf 𝑓)β€˜πΉ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆͺ cuni 4908  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  TopOnctopon 22411   Cn ccn 22727   CnP ccnp 22728  Filcfil 23348   fLim cflim 23437   fLimf cflf 23438
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 7724
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 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-map 8821  df-topgen 17388  df-fbas 20940  df-fg 20941  df-top 22395  df-topon 22412  df-ntr 22523  df-nei 22601  df-cn 22730  df-cnp 22731  df-fil 23349  df-fm 23441  df-flim 23442  df-flf 23443
This theorem is referenced by:  cnflf2  23506  flfcntr  23546  fmcncfil  32906
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