Theorem cnflf 23950

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 23228	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
2		simplr 767	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋⟶𝑌)
3		cnpflf 23949	. . . . . . 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 1170	. . . . . 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 705	. . . . 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 3165	. . . 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 2725	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
8	7	flimelbas 23916	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥 ∈ ∪ 𝐽)
9		toponuni 22860	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
10	9	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝑋 = ∪ 𝐽)
11	10	eleq2d 2811	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽))
12	8, 11	imbitrrid 245	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥 ∈ 𝑋))
13	12	pm4.71rd 561	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝑥 ∈ (𝐽 fLim 𝑓) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ (𝐽 fLim 𝑓))))
14	13	imbi1d 340	. . . . . . . 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 449	. . . . . . . 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 286	. . . . . . 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 3163	. . . . . 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 3167	. . . . 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 3276	. . . . 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 286	. . . 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 281	. . 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 577	. 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 278	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 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3050  ∪ cuni 4909  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419  TopOnctopon 22856   Cn ccn 23172   CnP ccnp 23173  Filcfil 23793   fLim cflim 23882   fLimf cflf 23883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847  df-topgen 17428  df-fbas 21293  df-fg 21294  df-top 22840  df-topon 22857  df-ntr 22968  df-nei 23046  df-cn 23175  df-cnp 23176  df-fil 23794  df-fm 23886  df-flim 23887  df-flf 23888

This theorem is referenced by:  cnflf2  23951  flfcntr  23991  fmcncfil  33663