Theorem cnextval 22671

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 4851	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
2	1	oveq2d 7174	. . 3 ⊢ (𝑗 = 𝐽 → (∪ 𝑘 ↑pm ∪ 𝑗) = (∪ 𝑘 ↑pm ∪ 𝐽))
3		fveq2 6672	. . . . 5 ⊢ (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
4	3	fveq1d 6674	. . . 4 ⊢ (𝑗 = 𝐽 → ((cls‘𝑗)‘dom 𝑓) = ((cls‘𝐽)‘dom 𝑓))
5		fveq2 6672	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → (nei‘𝑗) = (nei‘𝐽))
6	5	fveq1d 6674	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
7	6	oveq1d 7173	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓) = (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))
8	7	oveq2d 7174	. . . . . 6 ⊢ (𝑗 = 𝐽 → (𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓)) = (𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)))
9	8	fveq1d 6674	. . . . 5 ⊢ (𝑗 = 𝐽 → ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓) = ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))
10	9	xpeq2d 5587	. . . 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 4949	. . 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 5153	. 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 4851	. . . 4 ⊢ (𝑘 = 𝐾 → ∪ 𝑘 = ∪ 𝐾)
14	13	oveq1d 7173	. . 3 ⊢ (𝑘 = 𝐾 → (∪ 𝑘 ↑pm ∪ 𝐽) = (∪ 𝐾 ↑pm ∪ 𝐽))
15		oveq1 7165	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)))
16	15	fveq1d 6674	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))
17	16	xpeq2d 5587	. . . 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 4950	. . 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 5153	. 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 22670	. 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 7191	. . 3 ⊢ (∪ 𝐾 ↑pm ∪ 𝐽) ∈ V
22	21	mptex 6988	. 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 7312	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 398   = wceq 1537   ∈ wcel 2114  {csn 4569  ∪ cuni 4840  ∪ ciun 4921   ↦ cmpt 5148   × cxp 5555  dom cdm 5557  ‘cfv 6357  (class class class)co 7158   ↑_pm cpm 8409   ↾_t crest 16696  Topctop 21503  clsccl 21628  neicnei 21707   fLimf cflf 22545  CnExtccnext 22669

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-cnext 22670

This theorem is referenced by:  cnextfval  22672