Theorem cnextf 24095

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 24093	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹))
8	1, 2, 3, 4, 7	syl22anc 838	. . 3 ⊢ (𝜑 → Fun ((𝐽CnExt𝐾)‘𝐹))
9		dfdm3 5912	. . . 4 ⊢ dom ((𝐽CnExt𝐾)‘𝐹) = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)}
10		simpl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜑)
11		cnextf.6	. . . . . . . . 9 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)
12	11	eleq2d 2830	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ 𝐶))
13	12	biimpar 477	. . . . . . 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 4376	. . . . . . . 8 ⊢ (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
16	14, 15	sylib 218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
17		haustop 23360	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
18	2, 17	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Top)
19	5, 6	cnextfval 24091	. . . . . . . . . . . . 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 838	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
21	20	eleq2d 2830	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
22		opeliunxp 5767	. . . . . . . . . . 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 287	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
24	23	exbidv 1920	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ ∃𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
25		19.42v 1953	. . . . . . . . 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 287	. . . . . . . 8 ⊢ (𝜑 → (∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ ∃𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
27	26	biimpar 477	. . . . . . 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 836	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹))
29	26	simprbda 498	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘𝐴))
30	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ 𝐶))
31	29, 30	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)) → 𝑥 ∈ 𝐶)
32	28, 31	impbida 800	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)))
33	32	eqabdv 2878	. . . 4 ⊢ (𝜑 → 𝐶 = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)})
34	9, 33	eqtr4id 2799	. . 3 ⊢ (𝜑 → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
35		df-fn 6576	. . 3 ⊢ (((𝐽CnExt𝐾)‘𝐹) Fn 𝐶 ↔ (Fun ((𝐽CnExt𝐾)‘𝐹) ∧ dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶))
36	8, 34, 35	sylanbrc 582	. 2 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) Fn 𝐶)
37	20	rneqd 5963	. . 3 ⊢ (𝜑 → ran ((𝐽CnExt𝐾)‘𝐹) = ran ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
38		rniun 6179	. . . 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 3492	. . . . . . . . 9 ⊢ 𝑥 ∈ V
40	39	snnz 4801	. . . . . . . 8 ⊢ {𝑥} ≠ ∅
41		rnxp 6201	. . . . . . . 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 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝑥 ∈ 𝐶)
44	6	toptopon 22944	. . . . . . . . . . 11 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵))
45	18, 44	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐵))
46	45	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐾 ∈ (TopOn‘𝐵))
47	5	toptopon 22944	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶))
48	1, 47	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐶))
49	48	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐽 ∈ (TopOn‘𝐶))
50	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶)
51		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶)
52		trnei 23921	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)))
53	52	biimpa 476	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
55	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
56		flfelbas 24023	. . . . . . . . . . 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 412	. . . . . . . . . 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 4014	. . . . . . . . 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 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ⊆ 𝐵)
60	43, 59	syldan 590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ⊆ 𝐵)
61	42, 60	eqsstrid 4057	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
62	61	ralrimiva 3152	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
63		iunss 5068	. . . . 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 234	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
65	38, 64	eqsstrid 4057	. . 3 ⊢ (𝜑 → ran ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
66	37, 65	eqsstrd 4047	. 2 ⊢ (𝜑 → ran ((𝐽CnExt𝐾)‘𝐹) ⊆ 𝐵)
67		df-f 6577	. 2 ⊢ (((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵 ↔ (((𝐽CnExt𝐾)‘𝐹) Fn 𝐶 ∧ ran ((𝐽CnExt𝐾)‘𝐹) ⊆ 𝐵))
68	36, 66, 67	sylanbrc 582	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717   ≠ wne 2946  ∀wral 3067   ⊆ wss 3976  ∅c0 4352  {csn 4648  ⟨cop 4654  ∪ cuni 4931  ∪ ciun 5015   × cxp 5698  dom cdm 5700  ran crn 5701  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↾_t crest 17480  Topctop 22920  TopOnctopon 22937  clsccl 23047  neicnei 23126  Hauscha 23337  Filcfil 23874   fLimf cflf 23964  CnExtccnext 24088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-map 8886  df-pm 8887  df-rest 17482  df-fbas 21384  df-fg 21385  df-top 22921  df-topon 22938  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-haus 23344  df-fil 23875  df-fm 23967  df-flim 23968  df-flf 23969  df-cnext 24089

This theorem is referenced by:  cnextcn  24096  cnextfres1  24097