Theorem cnextfval 23983

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 23982	. . 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 5850	. . . . 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 6667	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → dom 𝐹 = 𝐴)
7	4, 6	eqtrd 2766	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴)
8	7	fveq2d 6832	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) ∧ 𝑓 = 𝐹) → ((cls‘𝐽)‘dom 𝑓) = ((cls‘𝐽)‘𝐴))
9	7	oveq2d 7368	. . . . . 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 7368	. . . . 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 6835	. . . 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 5649	. . 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 4971	. 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 7679	. . . 4 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ V)
15	14	ad2antlr 727	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → ∪ 𝐾 ∈ V)
16		uniexg 7679	. . . 4 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
17	16	ad2antrr 726	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝑋)) → ∪ 𝐽 ∈ V)
18		eqid 2731	. . . . . 6 ⊢ 𝐴 = 𝐴
19		cnextfval.2	. . . . . 6 ⊢ 𝐵 = ∪ 𝐾
20	18, 19	feq23i 6651	. . . . 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 3959	. . . . 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 8775	. . 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 6841	. . . 4 ⊢ ((cls‘𝐽)‘𝐴) ∈ V
30		vsnex 5374	. . . . 5 ⊢ {𝑥} ∈ V
31		fvex 6841	. . . . 5 ⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ∈ V
32	30, 31	xpex 7692	. . . 4 ⊢ ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ∈ V
33	29, 32	iunex 7906	. . 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 6942	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 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3897  {csn 4575  ∪ cuni 4858  ∪ ciun 4941   ↦ cmpt 5174   × cxp 5617  dom cdm 5619  ⟶wf 6483  ‘cfv 6487  (class class class)co 7352   ↑_pm cpm 8757   ↾_t crest 17330  Topctop 22814  clsccl 22939  neicnei 23018   fLimf cflf 23856  CnExtccnext 23980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-pm 8759  df-cnext 23981

This theorem is referenced by:  cnextrel  23984  cnextfun  23985  cnextfvval  23986  cnextf  23987  cnextfres  23990