Theorem cnflf 23967

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:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹‘𝑥) ∈ ((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

Step	Hyp	Ref	Expression
1		cncnp 23245	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
2		simplr 769	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋⟶𝑌)
3		cnpflf 23966	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))))
4	3	ad4ant124 1175	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))))
5	2, 4	mpbirand 708	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
6	5	ralbidva 3159	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
7		eqid 2737	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
8	7	flimelbas 23933	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥 ∈ ∪ 𝐽)
9		toponuni 22879	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
10	9	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝑋 = ∪ 𝐽)
11	10	eleq2d 2823	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽))
12	8, 11	imbitrrid 246	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥 ∈ 𝑋))
13	12	pm4.71rd 562	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝑥 ∈ (𝐽 fLim 𝑓) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ (𝐽 fLim 𝑓))))
14	13	imbi1d 341	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
15		impexp 450	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)) ↔ (𝑥 ∈ 𝑋 → (𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
16	14, 15	bitrdi 287	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)) ↔ (𝑥 ∈ 𝑋 → (𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))))
17	16	ralbidv2 3157	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹) ↔ ∀𝑥 ∈ 𝑋 (𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
18	17	ralbidv 3161	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹) ↔ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ 𝑋 (𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
19		ralcom 3266	. . . . 5 ⊢ (∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ 𝑋 (𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))
20	18, 19	bitrdi 287	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹) ↔ ∀𝑥 ∈ 𝑋 ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
21	6, 20	bitr4d 282	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))
22	21	pm5.32da 579	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
23	1, 22	bitrd 279	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∪ cuni 4851  ⟶wf 6495  ‘cfv 6499  (class class class)co 7367  TopOnctopon 22875   Cn ccn 23189   CnP ccnp 23190  Filcfil 23810   fLim cflim 23899   fLimf cflf 23900

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-topgen 17406  df-fbas 21349  df-fg 21350  df-top 22859  df-topon 22876  df-ntr 22985  df-nei 23063  df-cn 23192  df-cnp 23193  df-fil 23811  df-fm 23903  df-flim 23904  df-flf 23905

This theorem is referenced by:  cnflf2  23968  flfcntr  24008  fmcncfil  34075