Theorem cnextval 24036

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 4862	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
2	1	oveq2d 7376	. . 3 ⊢ (𝑗 = 𝐽 → (∪ 𝑘 ↑pm ∪ 𝑗) = (∪ 𝑘 ↑pm ∪ 𝐽))
3		fveq2 6834	. . . . 5 ⊢ (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
4	3	fveq1d 6836	. . . 4 ⊢ (𝑗 = 𝐽 → ((cls‘𝑗)‘dom 𝑓) = ((cls‘𝐽)‘dom 𝑓))
5		fveq2 6834	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (nei‘𝑗) = (nei‘𝐽))
6	5	fveq1d 6836	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
7	6	oveq1d 7375	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓) = (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))
8	7	oveq2d 7376	. . . . . 6 ⊢ (𝑗 = 𝐽 → (𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓)) = (𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)))
9	8	fveq1d 6836	. . . . 5 ⊢ (𝑗 = 𝐽 → ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓) = ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))
10	9	xpeq2d 5654	. . . 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 4964	. . 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 5173	. 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 4862	. . . 4 ⊢ (𝑘 = 𝐾 → ∪ 𝑘 = ∪ 𝐾)
14	13	oveq1d 7375	. . 3 ⊢ (𝑘 = 𝐾 → (∪ 𝑘 ↑pm ∪ 𝐽) = (∪ 𝐾 ↑pm ∪ 𝐽))
15		oveq1 7367	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)))
16	15	fveq1d 6836	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))
17	16	xpeq2d 5654	. . . 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 4965	. . 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 5173	. 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 24035	. 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 7393	. . 3 ⊢ (∪ 𝐾 ↑pm ∪ 𝐽) ∈ V
22	21	mptex 7171	. 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 7520	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 1542   ∈ wcel 2114  {csn 4568  ∪ cuni 4851  ∪ ciun 4934   ↦ cmpt 5167   × cxp 5622  dom cdm 5624  ‘cfv 6492  (class class class)co 7360   ↑_pm cpm 8767   ↾_t crest 17374  Topctop 22868  clsccl 22993  neicnei 23072   fLimf cflf 23910  CnExtccnext 24034

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-pr 5370

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-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-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-cnext 24035

This theorem is referenced by:  cnextfval  24037