Theorem cnextfval 23955

Description: The continuous extension of a given function 𝐹. (Contributed by Thierry Arnoux, 1-Dec-2017.)

Hypotheses

Ref	Expression
cnextfval.1	⊢ 𝑋 = ∪ 𝐽
cnextfval.2	⊢ 𝐵 = ∪ 𝐾

Assertion

Ref	Expression
cnextfval	⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))

Distinct variable groups:   𝑥,𝐽   𝑥,𝐾   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝑋

Proof of Theorem cnextfval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnextval 23954	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽CnExt𝐾) = (𝑓 ∈ (∪ 𝐾 ↑pm ∪ 𝐽) ↦ ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
2	1	adantr 480	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → (𝐽CnExt𝐾) = (𝑓 ∈ (∪ 𝐾 ↑pm ∪ 𝐽) ↦ ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
3		simpr 484	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
4	3	dmeqd 5871	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
5		simplrl 776	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → 𝐹:𝐴⟶𝐵)
6	5	fdmd 6700	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → dom 𝐹 = 𝐴)
7	4, 6	eqtrd 2765	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴)
8	7	fveq2d 6864	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → ((cls‘𝐽)‘dom 𝑓) = ((cls‘𝐽)‘𝐴))
9	7	oveq2d 7405	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓) = (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))
10	9	oveq2d 7405	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)))
11	10, 3	fveq12d 6867	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
12	11	xpeq2d 5670	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)) = ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
13	8, 12	iuneq12d 4987	. 2 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
14		uniexg 7718	. . . 4 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ V)
15	14	ad2antlr 727	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → ∪ 𝐾 ∈ V)
16		uniexg 7718	. . . 4 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
17	16	ad2antrr 726	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → ∪ 𝐽 ∈ V)
18		eqid 2730	. . . . . 6 ⊢ 𝐴 = 𝐴
19		cnextfval.2	. . . . . 6 ⊢ 𝐵 = ∪ 𝐾
20	18, 19	feq23i 6684	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶∪ 𝐾)
21	20	biimpi 216	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶∪ 𝐾)
22	21	ad2antrl 728	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → 𝐹:𝐴⟶∪ 𝐾)
23		cnextfval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
24	23	sseq2i 3978	. . . . 5 ⊢ (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ 𝐽)
25	24	biimpi 216	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → 𝐴 ⊆ ∪ 𝐽)
26	25	ad2antll 729	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → 𝐴 ⊆ ∪ 𝐽)
27		elpm2r 8820	. . 3 ⊢ (((∪ 𝐾 ∈ V ∧ ∪ 𝐽 ∈ V) ∧ (𝐹:𝐴⟶∪ 𝐾 ∧ 𝐴 ⊆ ∪ 𝐽)) → 𝐹 ∈ (∪ 𝐾 ↑pm ∪ 𝐽))
28	15, 17, 22, 26, 27	syl22anc 838	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → 𝐹 ∈ (∪ 𝐾 ↑pm ∪ 𝐽))
29		fvex 6873	. . . 4 ⊢ ((cls‘𝐽)‘𝐴) ∈ V
30		vsnex 5391	. . . . 5 ⊢ {𝑥} ∈ V
31		fvex 6873	. . . . 5 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ∈ V
32	30, 31	xpex 7731	. . . 4 ⊢ ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ∈ V
33	29, 32	iunex 7949	. . 3 ⊢ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ∈ V
34	33	a1i 11	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ∈ V)
35	2, 13, 28, 34	fvmptd 6977	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → ((𝐽CnExt𝐾)‘𝐹) = ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3916  {csn 4591  ∪ cuni 4873  ∪ ciun 4957   ↦ cmpt 5190   × cxp 5638  dom cdm 5640  ⟶wf 6509  ‘cfv 6513  (class class class)co 7389   ↑_pm cpm 8802   ↾_t crest 17389  Topctop 22786  clsccl 22911  neicnei 22990   fLimf cflf 23828  CnExtccnext 23952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-pm 8804  df-cnext 23953

This theorem is referenced by:  cnextrel  23956  cnextfun  23957  cnextfvval  23958  cnextf  23959  cnextfres  23962