Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnflf Structured version   Visualization version   GIF version

Theorem cnflf 22616
 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 21894 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
2 simplr 768 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) ∧ 𝑥𝑋) → 𝐹:𝑋𝑌)
3 cnpflf 22615 . . . . . . 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 1170 . . . . . 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 706 . . . . 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 3191 . . . 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 2824 . . . . . . . . . . . 12 𝐽 = 𝐽
87flimelbas 22582 . . . . . . . . . . 11 (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥 𝐽)
9 toponuni 21528 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
109ad2antrr 725 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → 𝑋 = 𝐽)
1110eleq2d 2901 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥𝑋𝑥 𝐽))
128, 11syl5ibr 249 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥𝑋))
1312pm4.71rd 566 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋𝑌) → (𝑥 ∈ (𝐽 fLim 𝑓) ↔ (𝑥𝑋𝑥 ∈ (𝐽 fLim 𝑓))))
1413imbi1d 345 . . . . . . . 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 454 . . . . . . . 8 (((𝑥𝑋𝑥 ∈ (𝐽 fLim 𝑓)) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)) ↔ (𝑥𝑋 → (𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
1614, 15syl6bb 290 . . . . . . 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 3190 . . . . . 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 3192 . . . . 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 3345 . . . . 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, 19syl6bb 290 . . . 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 285 . . 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 582 . 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 282 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3133  ∪ cuni 4824  ⟶wf 6341  ‘cfv 6345  (class class class)co 7151  TopOnctopon 21524   Cn ccn 21838   CnP ccnp 21839  Filcfil 22459   fLim cflim 22548   fLimf cflf 22549 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7686  df-2nd 7687  df-map 8406  df-topgen 16719  df-fbas 20097  df-fg 20098  df-top 21508  df-topon 21525  df-ntr 21634  df-nei 21712  df-cn 21841  df-cnp 21842  df-fil 22460  df-fm 22552  df-flim 22553  df-flf 22554 This theorem is referenced by:  cnflf2  22617  flfcntr  22657  fmcncfil  31259
 Copyright terms: Public domain W3C validator