Theorem cnextf 24040

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 24038	. . . 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 5834	. . . 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 2823	. . . . . . . 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 4294	. . . . . . . 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 23305	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
18	2, 17	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Top)
19	5, 6	cnextfval 24036	. . . . . . . . . . . . 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 5689	. . . . . . . . . . 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 1923	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ ∃𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
25		19.42v 1955	. . . . . . . . 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 2870	. . . 4 ⊢ (𝜑 → 𝐶 = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((𝐽CnExt𝐾)‘𝐹)})
34	9, 33	eqtr4id 2791	. . 3 ⊢ (𝜑 → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶)
35		df-fn 6493	. . 3 ⊢ (((𝐽CnExt𝐾)‘𝐹) Fn 𝐶 ↔ (Fun ((𝐽CnExt𝐾)‘𝐹) ∧ dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶))
36	8, 34, 35	sylanbrc 584	. 2 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) Fn 𝐶)
37	20	rneqd 5885	. . 3 ⊢ (𝜑 → ran ((𝐽CnExt𝐾)‘𝐹) = ran ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
38		rniun 6103	. . . 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 3434	. . . . . . . . 9 ⊢ 𝑥 ∈ V
40	39	snnz 4721	. . . . . . . 8 ⊢ {𝑥} ≠ ∅
41		rnxp 6126	. . . . . . . 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 22891	. . . . . . . . . . 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 22891	. . . . . . . . . . . 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 23866	. . . . . . . . . . 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 1376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))
55	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐹:𝐴⟶𝐵)
56		flfelbas 23968	. . . . . . . . . . 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 3928	. . . . . . . . 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 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ⊆ 𝐵)
60	43, 59	syldan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ⊆ 𝐵)
61	42, 60	eqsstrid 3961	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
62	61	ralrimiva 3130	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ((cls‘𝐽)‘𝐴)ran ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
63		iunss 4988	. . . . 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 3961	. . 3 ⊢ (𝜑 → ran ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ⊆ 𝐵)
66	37, 65	eqsstrd 3957	. 2 ⊢ (𝜑 → ran ((𝐽CnExt𝐾)‘𝐹) ⊆ 𝐵)
67		df-f 6494	. 2 ⊢ (((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵 ↔ (((𝐽CnExt𝐾)‘𝐹) Fn 𝐶 ∧ ran ((𝐽CnExt𝐾)‘𝐹) ⊆ 𝐵))
68	36, 66, 67	sylanbrc 584	1 ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052   ⊆ wss 3890  ∅c0 4274  {csn 4568  ⟨cop 4574  ∪ cuni 4851  ∪ ciun 4934   × cxp 5620  dom cdm 5622  ran crn 5623  Fun wfun 6484   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7358   ↾_t crest 17372  Topctop 22867  TopOnctopon 22884  clsccl 22992  neicnei 23071  Hauscha 23282  Filcfil 23819   fLimf cflf 23909  CnExtccnext 24033

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

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-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8766  df-pm 8767  df-rest 17374  df-fbas 21339  df-fg 21340  df-top 22868  df-topon 22885  df-cld 22993  df-ntr 22994  df-cls 22995  df-nei 23072  df-haus 23289  df-fil 23820  df-fm 23912  df-flim 23913  df-flf 23914  df-cnext 24034

This theorem is referenced by:  cnextcn  24041  cnextfres1  24042