Theorem cnextf 22963

Description: Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)

Hypotheses

Ref	Expression
cnextf.1	⊢ 𝐶 = ∪ 𝐽
cnextf.2	⊢ 𝐵 = ∪ 𝐾
cnextf.3	⊢ (𝜑 → 𝐽 ∈ Top)
cnextf.4	⊢ (𝜑 → 𝐾 ∈ Haus)
cnextf.5	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
cnextf.a	⊢ (𝜑 → 𝐴 ⊆ 𝐶)
cnextf.6	⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)
cnextf.7	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)

Assertion

Ref	Expression
cnextf	⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝜑,𝑥

Proof of Theorem cnextf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnextf.3	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Top)
2		cnextf.4	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Haus)
3		cnextf.5	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4		cnextf.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
5		cnextf.1	. . . . 5 ⊢ 𝐶 = ∪ 𝐽
6		cnextf.2	. . . . 5 ⊢ 𝐵 = ∪ 𝐾
7	5, 6	cnextfun 22961	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹))
8	1, 2, 3, 4, 7	syl22anc 839	. . 3 ⊢ (𝜑 → Fun ((𝐽CnExt𝐾)‘𝐹))
9		dfdm3 5756	. . . 4 ⊢ dom ((𝐽CnExt𝐾)‘𝐹) = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)}
10		simpl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜑)
11		cnextf.6	. . . . . . . . 9 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)
12	11	eleq2d 2823	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ 𝐶))
13	12	biimpar 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
14		cnextf.7	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)
15		n0 4261	. . . . . . . 8 ⊢ (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
16	14, 15	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
17		haustop 22228	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
18	2, 17	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Top)
19	5, 6	cnextfval 22959	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
20	1, 18, 3, 4, 19	syl22anc 839	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
21	20	eleq2d 2823	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
22		opeliunxp 5616	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
23	21, 22	bitrdi 290	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
24	23	exbidv 1929	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ ∃𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
25		19.42v 1962	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
26	24, 25	bitrdi 290	. . . . . . . 8 ⊢ (𝜑 → (∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
27	26	biimpar 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) → ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹))
28	10, 13, 16, 27	syl12anc 837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹))
29	26	simprbda 502	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
30	12	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ 𝐶))
31	29, 30	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)) → 𝑥 ∈ 𝐶)
32	28, 31	impbida 801	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)))
33	32	abbi2dv 2874	. . . 4 ⊢ (𝜑 → 𝐶 = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)})
34	9, 33	eqtr4id 2797	. . 3 ⊢ (𝜑 → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
35		df-fn 6383	. . 3 ⊢ (((𝐽CnExt𝐾)‘𝐹) Fn 𝐶 ↔ (Fun ((𝐽CnExt𝐾)‘𝐹) ∧ dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶))
36	8, 34, 35	sylanbrc 586	. 2 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) Fn 𝐶)
37	20	rneqd 5807	. . 3 ⊢ (𝜑 → ran ((𝐽CnExt𝐾)‘𝐹) = ran ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
38		rniun 6011	. . . 4 ⊢ ran ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
39		vex 3412	. . . . . . . . 9 ⊢ 𝑥 ∈ V
40	39	snnz 4692	. . . . . . . 8 ⊢ {𝑥} ≠ ∅
41		rnxp 6033	. . . . . . . 8 ⊢ ({𝑥} ≠ ∅ → ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)
43	12	biimpa 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝑥 ∈ 𝐶)
44	6	toptopon 21814	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
45	18, 44	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐵))
46	45	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐾 ∈ (TopOn‘𝐵))
47	5	toptopon 21814	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
48	1, 47	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐶))
49	48	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐽 ∈ (TopOn‘𝐶))
50	4	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
51		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
52		trnei 22789	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
53	52	biimpa 480	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐶) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
54	49, 50, 51, 13, 53	syl31anc 1375	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
55	3	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
56		flfelbas 22891	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ (TopOn‘𝐵) ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) → 𝑦 ∈ 𝐵)
57	56	ex 416	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝐵) ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝐵) → (𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) → 𝑦 ∈ 𝐵))
58	57	ssrdv 3907	. . . . . . . . 9 ⊢ ((𝐾 ∈ (TopOn‘𝐵) ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝐵) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ⊆ 𝐵)
59	46, 54, 55, 58	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ⊆ 𝐵)
60	43, 59	syldan 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ⊆ 𝐵)
61	42, 60	eqsstrid 3949	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
62	61	ralrimiva 3105	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
63		iunss 4954	. . . . 5 ⊢ (∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵 ↔ ∀𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
64	62, 63	sylibr 237	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
65	38, 64	eqsstrid 3949	. . 3 ⊢ (𝜑 → ran ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
66	37, 65	eqsstrd 3939	. 2 ⊢ (𝜑 → ran ((𝐽CnExt𝐾)‘𝐹) ⊆ 𝐵)
67		df-f 6384	. 2 ⊢ (((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵 ↔ (((𝐽CnExt𝐾)‘𝐹) Fn 𝐶 ∧ ran ((𝐽CnExt𝐾)‘𝐹) ⊆ 𝐵))
68	36, 66, 67	sylanbrc 586	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543  ∃wex 1787   ∈ wcel 2110  {cab 2714   ≠ wne 2940  ∀wral 3061   ⊆ wss 3866  ∅c0 4237  {csn 4541  ⟨cop 4547  ∪ cuni 4819  ∪ ciun 4904   × cxp 5549  dom cdm 5551  ran crn 5552  Fun wfun 6374   Fn wfn 6375  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213   ↾_t crest 16925  Topctop 21790  TopOnctopon 21807  clsccl 21915  neicnei 21994  Hauscha 22205  Filcfil 22742   fLimf cflf 22832  CnExtccnext 22956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-map 8510  df-pm 8511  df-rest 16927  df-fbas 20360  df-fg 20361  df-top 21791  df-topon 21808  df-cld 21916  df-ntr 21917  df-cls 21918  df-nei 21995  df-haus 22212  df-fil 22743  df-fm 22835  df-flim 22836  df-flf 22837  df-cnext 22957

This theorem is referenced by:  cnextcn  22964  cnextfres1  22965