Theorem cnflf 23938

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 23216	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
2		simplr 768	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋⟶𝑌)
3		cnpflf 23937	. . . . . . 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 1174	. . . . . 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 707	. . . . 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 3161	. . . 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 2735	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
8	7	flimelbas 23904	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥 ∈ ∪ 𝐽)
9		toponuni 22850	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
10	9	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝑋 = ∪ 𝐽)
11	10	eleq2d 2820	. . . . . . . . . . 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 3159	. . . . . 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 3163	. . . . 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 3270	. . . . 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 1540   ∈ wcel 2108  ∀wral 3051  ∪ cuni 4883  ⟶wf 6526  ‘cfv 6530  (class class class)co 7403  TopOnctopon 22846   Cn ccn 23160   CnP ccnp 23161  Filcfil 23781   fLim cflim 23870   fLimf cflf 23871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7986  df-2nd 7987  df-map 8840  df-topgen 17455  df-fbas 21310  df-fg 21311  df-top 22830  df-topon 22847  df-ntr 22956  df-nei 23034  df-cn 23163  df-cnp 23164  df-fil 23782  df-fm 23874  df-flim 23875  df-flf 23876

This theorem is referenced by:  cnflf2  23939  flfcntr  23979  fmcncfil  33908