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Theorem cnflf 24026
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 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
Distinct variable groups:   𝑥,𝑓,𝑋   𝑓,𝑌,𝑥   𝑓,𝐹,𝑥   𝑓,𝐽,𝑥   𝑓,𝐾,𝑥

Proof of Theorem cnflf
StepHypRef Expression
1 cncnp 23304 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
2 simplr 769 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝑋) → 𝐹:𝑋𝑌)
3 cnpflf 24025 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))))
43ad4ant124 1172 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))))
52, 4mpbirand 707 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
65ralbidva 3174 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑥𝑋𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
7 eqid 2735 . . . . . . . . . . . 12 𝐽 = 𝐽
87flimelbas 23992 . . . . . . . . . . 11 (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥 𝐽)
9 toponuni 22936 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
109ad2antrr 726 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝑋 = 𝐽)
1110eleq2d 2825 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥𝑋𝑥 𝐽))
128, 11imbitrrid 246 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥𝑋))
1312pm4.71rd 562 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥 ∈ (𝐽 fLim 𝑓) ↔ (𝑥𝑋𝑥 ∈ (𝐽 fLim 𝑓))))
1413imbi1d 341 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)) ↔ ((𝑥𝑋𝑥 ∈ (𝐽 fLim 𝑓)) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
15 impexp 450 . . . . . . . 8 (((𝑥𝑋𝑥 ∈ (𝐽 fLim 𝑓)) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)) ↔ (𝑥𝑋 → (𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
1614, 15bitrdi 287 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → ((𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)) ↔ (𝑥𝑋 → (𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))))
1716ralbidv2 3172 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹) ↔ ∀𝑥𝑋 (𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
1817ralbidv 3176 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹) ↔ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥𝑋 (𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
19 ralcom 3287 . . . . 5 (∀𝑓 ∈ (Fil‘𝑋)∀𝑥𝑋 (𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)) ↔ ∀𝑥𝑋𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))
2018, 19bitrdi 287 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹) ↔ ∀𝑥𝑋𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
216, 20bitr4d 282 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))
2221pm5.32da 579 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
231, 22bitrd 279 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059   cuni 4912  wf 6559  cfv 6563  (class class class)co 7431  TopOnctopon 22932   Cn ccn 23248   CnP ccnp 23249  Filcfil 23869   fLim cflim 23958   fLimf cflf 23959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-topgen 17490  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-ntr 23044  df-nei 23122  df-cn 23251  df-cnp 23252  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964
This theorem is referenced by:  cnflf2  24027  flfcntr  24067  fmcncfil  33892
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