Theorem cnextf 24090

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 24088	. . . 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 5901	. . . 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 2825	. . . . . . . 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 4359	. . . . . . . 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 23355	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
18	2, 17	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Top)
19	5, 6	cnextfval 24086	. . . . . . . . . . . . 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 2825	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
22		opeliunxp 5756	. . . . . . . . . . 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 1919	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ ∃𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
25		19.42v 1951	. . . . . . . . 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 837	. . . . . 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 801	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)))
33	32	eqabdv 2873	. . . 4 ⊢ (𝜑 → 𝐶 = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)})
34	9, 33	eqtr4id 2794	. . 3 ⊢ (𝜑 → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
35		df-fn 6566	. . 3 ⊢ (((𝐽CnExt𝐾)‘𝐹) Fn 𝐶 ↔ (Fun ((𝐽CnExt𝐾)‘𝐹) ∧ dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶))
36	8, 34, 35	sylanbrc 583	. 2 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) Fn 𝐶)
37	20	rneqd 5952	. . 3 ⊢ (𝜑 → ran ((𝐽CnExt𝐾)‘𝐹) = ran ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
38		rniun 6170	. . . 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 3482	. . . . . . . . 9 ⊢ 𝑥 ∈ V
40	39	snnz 4781	. . . . . . . 8 ⊢ {𝑥} ≠ ∅
41		rnxp 6192	. . . . . . . 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 22939	. . . . . . . . . . 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 22939	. . . . . . . . . . . 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 23916	. . . . . . . . . . 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 1372	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
55	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
56		flfelbas 24018	. . . . . . . . . . 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 4001	. . . . . . . . 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 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ⊆ 𝐵)
60	43, 59	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ⊆ 𝐵)
61	42, 60	eqsstrid 4044	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
62	61	ralrimiva 3144	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
63		iunss 5050	. . . . 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 4044	. . 3 ⊢ (𝜑 → ran ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
66	37, 65	eqsstrd 4034	. 2 ⊢ (𝜑 → ran ((𝐽CnExt𝐾)‘𝐹) ⊆ 𝐵)
67		df-f 6567	. 2 ⊢ (((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵 ↔ (((𝐽CnExt𝐾)‘𝐹) Fn 𝐶 ∧ ran ((𝐽CnExt𝐾)‘𝐹) ⊆ 𝐵))
68	36, 66, 67	sylanbrc 583	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712   ≠ wne 2938  ∀wral 3059   ⊆ wss 3963  ∅c0 4339  {csn 4631  ⟨cop 4637  ∪ cuni 4912  ∪ ciun 4996   × cxp 5687  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ↾_t crest 17467  Topctop 22915  TopOnctopon 22932  clsccl 23042  neicnei 23121  Hauscha 23332  Filcfil 23869   fLimf cflf 23959  CnExtccnext 24083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-pm 8868  df-rest 17469  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-haus 23339  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-cnext 24084

This theorem is referenced by:  cnextcn  24091  cnextfres1  24092