Theorem cnpflf 23980

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 23229	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1119	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
3	2	3adantl3 1170	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
4		cnpflfi 23978	. . . . . . 7 ⊢ ((𝐴 ∈ (𝐽 fLim 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))
5	4	expcom 413	. . . . . 6 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))
6	5	ralrimivw 3134	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))
7	6	adantl 481	. . . 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 511	. . 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 412	. 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 1193	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋))
11		simpl3 1195	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ 𝑋)
12		neiflim 23953	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
13	10, 11, 12	syl2anc 585	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
14	11	snssd 4753	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ⊆ 𝑋)
15	11	snn0d 4720	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ≠ ∅)
16		neifil 23859	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
17	10, 14, 15, 16	syl3anc 1374	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
18		oveq2 7370	. . . . . . . . 9 ⊢ (𝑓 = ((nei‘𝐽)‘{𝐴}) → (𝐽 fLim 𝑓) = (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
19	18	eleq2d 2823	. . . . . . . 8 ⊢ (𝑓 = ((nei‘𝐽)‘{𝐴}) → (𝐴 ∈ (𝐽 fLim 𝑓) ↔ 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴}))))
20		oveq2 7370	. . . . . . . . . 10 ⊢ (𝑓 = ((nei‘𝐽)‘{𝐴}) → (𝐾 fLimf 𝑓) = (𝐾 fLimf ((nei‘𝐽)‘{𝐴})))
21	20	fveq1d 6838	. . . . . . . . 9 ⊢ (𝑓 = ((nei‘𝐽)‘{𝐴}) → ((𝐾 fLimf 𝑓)‘𝐹) = ((𝐾 fLimf ((nei‘𝐽)‘{𝐴}))‘𝐹))
22	21	eleq2d 2823	. . . . . . . 8 ⊢ (𝑓 = ((nei‘𝐽)‘{𝐴}) → ((𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹) ↔ (𝐹‘𝐴) ∈ ((𝐾 fLimf ((nei‘𝐽)‘{𝐴}))‘𝐹)))
23	19, 22	imbi12d 344	. . . . . . 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))))
24	23	rspcv 3561	. . . . . 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))))
25	17, 24	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))))
26	13, 25	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)))
27	26	imdistanda 571	. . 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))))
28		eqid 2737	. . . 4 ⊢ ((nei‘𝐽)‘{𝐴}) = ((nei‘𝐽)‘{𝐴})
29	28	cnpflf2 23979	. . 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))))
30	27, 29	sylibrd 259	. 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)))
31	9, 30	impbid 212	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ⊆ wss 3890  ∅c0 4274  {csn 4568  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362  TopOnctopon 22889  neicnei 23076   CnP ccnp 23204  Filcfil 23824   fLim cflim 23913   fLimf cflf 23914

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 5304  ax-pr 5372  ax-un 7684

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 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7937  df-2nd 7938  df-map 8770  df-fbas 21345  df-fg 21346  df-top 22873  df-topon 22890  df-ntr 22999  df-nei 23077  df-cnp 23207  df-fil 23825  df-fm 23917  df-flim 23918  df-flf 23919

This theorem is referenced by:  cnflf  23981  cnpfcf  24020