Theorem cnpflf 22175

Description: Continuity of a function at a point in terms of filter limits. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)

Assertion

Ref	Expression
cnpflf	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))))

Distinct variable groups:   𝐴,𝑓   𝑓,𝑋   𝑓,𝑌   𝑓,𝐹   𝑓,𝐽   𝑓,𝐾

Proof of Theorem cnpflf

Step	Hyp	Ref	Expression
1		cnpf2 21425	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1153	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
3	2	3adantl3 1215	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
4		cnpflfi 22173	. . . . . . 7 ⊢ ((𝐴 ∈ (𝐽 fLim 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))
5	4	expcom 404	. . . . . 6 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))
6	5	ralrimivw 3176	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))
7	6	adantl 475	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))
8	3, 7	jca 509	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
9	8	ex 403	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))))
10		simpl1 1248	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋))
11		simpl3 1252	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ 𝑋)
12		neiflim 22148	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
13	10, 11, 12	syl2anc 581	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
14	11	snssd 4558	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ⊆ 𝑋)
15		snnzg 4527	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ≠ ∅)
16	11, 15	syl 17	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ≠ ∅)
17		neifil 22054	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
18	10, 14, 16, 17	syl3anc 1496	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
19		oveq2 6913	. . . . . . . . 9 ⊢ (𝑓 = ((nei‘𝐽)‘{𝐴}) → (𝐽 fLim 𝑓) = (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
20	19	eleq2d 2892	. . . . . . . 8 ⊢ (𝑓 = ((nei‘𝐽)‘{𝐴}) → (𝐴 ∈ (𝐽 fLim 𝑓) ↔ 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴}))))
21		oveq2 6913	. . . . . . . . . 10 ⊢ (𝑓 = ((nei‘𝐽)‘{𝐴}) → (𝐾 fLimf 𝑓) = (𝐾 fLimf ((nei‘𝐽)‘{𝐴})))
22	21	fveq1d 6435	. . . . . . . . 9 ⊢ (𝑓 = ((nei‘𝐽)‘{𝐴}) → ((𝐾 fLimf 𝑓)‘𝐹) = ((𝐾 fLimf ((nei‘𝐽)‘{𝐴}))‘𝐹))
23	22	eleq2d 2892	. . . . . . . 8 ⊢ (𝑓 = ((nei‘𝐽)‘{𝐴}) → ((𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹) ↔ (𝐹‘𝐴) ∈ ((𝐾 fLimf ((nei‘𝐽)‘{𝐴}))‘𝐹)))
24	20, 23	imbi12d 336	. . . . . . 7 ⊢ (𝑓 = ((nei‘𝐽)‘{𝐴}) → ((𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)) ↔ (𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})) → (𝐹‘𝐴) ∈ ((𝐾 fLimf ((nei‘𝐽)‘{𝐴}))‘𝐹))))
25	24	rspcv 3522	. . . . . 6 ⊢ (((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)) → (𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})) → (𝐹‘𝐴) ∈ ((𝐾 fLimf ((nei‘𝐽)‘{𝐴}))‘𝐹))))
26	18, 25	syl 17	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)) → (𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})) → (𝐹‘𝐴) ∈ ((𝐾 fLimf ((nei‘𝐽)‘{𝐴}))‘𝐹))))
27	13, 26	mpid 44	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf ((nei‘𝐽)‘{𝐴}))‘𝐹)))
28	27	imdistanda 569	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))) → (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝐴) ∈ ((𝐾 fLimf ((nei‘𝐽)‘{𝐴}))‘𝐹))))
29		eqid 2825	. . . 4 ⊢ ((nei‘𝐽)‘{𝐴}) = ((nei‘𝐽)‘{𝐴})
30	29	cnpflf2 22174	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝐴) ∈ ((𝐾 fLimf ((nei‘𝐽)‘{𝐴}))‘𝐹))))
31	28, 30	sylibrd 251	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)))
32	9, 31	impbid 204	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166   ≠ wne 2999  ∀wral 3117   ⊆ wss 3798  ∅c0 4144  {csn 4397  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  TopOnctopon 21085  neicnei 21272   CnP ccnp 21400  Filcfil 22019   fLim cflim 22108   fLimf cflf 22109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-map 8124  df-fbas 20103  df-fg 20104  df-top 21069  df-topon 21086  df-ntr 21195  df-nei 21273  df-cnp 21403  df-fil 22020  df-fm 22112  df-flim 22113  df-flf 22114

This theorem is referenced by:  cnflf  22176  cnpfcf  22215