Theorem cnextval 24070

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

Assertion

Ref	Expression
cnextval	⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽CnExt𝐾) = (𝑓 ∈ (∪ 𝐾 ↑pm ∪ 𝐽) ↦ ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))

Distinct variable groups:   𝑥,𝑓,𝐽   𝑓,𝐾,𝑥

Proof of Theorem cnextval

Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4917	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
2	1	oveq2d 7448	. . 3 ⊢ (𝑗 = 𝐽 → (∪ 𝑘 ↑pm ∪ 𝑗) = (∪ 𝑘 ↑pm ∪ 𝐽))
3		fveq2 6905	. . . . 5 ⊢ (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
4	3	fveq1d 6907	. . . 4 ⊢ (𝑗 = 𝐽 → ((cls‘𝑗)‘dom 𝑓) = ((cls‘𝐽)‘dom 𝑓))
5		fveq2 6905	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (nei‘𝑗) = (nei‘𝐽))
6	5	fveq1d 6907	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
7	6	oveq1d 7447	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓) = (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))
8	7	oveq2d 7448	. . . . . 6 ⊢ (𝑗 = 𝐽 → (𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓)) = (𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)))
9	8	fveq1d 6907	. . . . 5 ⊢ (𝑗 = 𝐽 → ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓) = ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))
10	9	xpeq2d 5714	. . . 4 ⊢ (𝑗 = 𝐽 → ({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓)) = ({𝑥} × ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)))
11	4, 10	iuneq12d 5020	. . 3 ⊢ (𝑗 = 𝐽 → ∪ 𝑥 ∈ ((cls‘𝑗)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓)) = ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)))
12	2, 11	mpteq12dv 5232	. 2 ⊢ (𝑗 = 𝐽 → (𝑓 ∈ (∪ 𝑘 ↑pm ∪ 𝑗) ↦ ∪ 𝑥 ∈ ((cls‘𝑗)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓))) = (𝑓 ∈ (∪ 𝑘 ↑pm ∪ 𝐽) ↦ ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
13		unieq 4917	. . . 4 ⊢ (𝑘 = 𝐾 → ∪ 𝑘 = ∪ 𝐾)
14	13	oveq1d 7447	. . 3 ⊢ (𝑘 = 𝐾 → (∪ 𝑘 ↑pm ∪ 𝐽) = (∪ 𝐾 ↑pm ∪ 𝐽))
15		oveq1 7439	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)))
16	15	fveq1d 6907	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))
17	16	xpeq2d 5714	. . . 4 ⊢ (𝑘 = 𝐾 → ({𝑥} × ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)) = ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)))
18	17	iuneq2d 5021	. . 3 ⊢ (𝑘 = 𝐾 → ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)) = ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)))
19	14, 18	mpteq12dv 5232	. 2 ⊢ (𝑘 = 𝐾 → (𝑓 ∈ (∪ 𝑘 ↑pm ∪ 𝐽) ↦ ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))) = (𝑓 ∈ (∪ 𝐾 ↑pm ∪ 𝐽) ↦ ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
20		df-cnext 24069	. 2 ⊢ CnExt = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ (∪ 𝑘 ↑pm ∪ 𝑗) ↦ ∪ 𝑥 ∈ ((cls‘𝑗)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
21		ovex 7465	. . 3 ⊢ (∪ 𝐾 ↑pm ∪ 𝐽) ∈ V
22	21	mptex 7244	. 2 ⊢ (𝑓 ∈ (∪ 𝐾 ↑pm ∪ 𝐽) ↦ ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))) ∈ V
23	12, 19, 20, 22	ovmpo 7594	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽CnExt𝐾) = (𝑓 ∈ (∪ 𝐾 ↑pm ∪ 𝐽) ↦ ∪ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {csn 4625  ∪ cuni 4906  ∪ ciun 4990   ↦ cmpt 5224   × cxp 5682  dom cdm 5684  ‘cfv 6560  (class class class)co 7432   ↑_pm cpm 8868   ↾_t crest 17466  Topctop 22900  clsccl 23027  neicnei 23106   fLimf cflf 23944  CnExtccnext 24068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-cnext 24069

This theorem is referenced by:  cnextfval  24071