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Theorem cnextval 21675
 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 ∈ ( 𝐾pm 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.
StepHypRef Expression
1 unieq 4380 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
21oveq2d 6565 . . 3 (𝑗 = 𝐽 → ( 𝑘pm 𝑗) = ( 𝑘pm 𝐽))
3 fveq2 6103 . . . . 5 (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
43fveq1d 6105 . . . 4 (𝑗 = 𝐽 → ((cls‘𝑗)‘dom 𝑓) = ((cls‘𝐽)‘dom 𝑓))
5 fveq2 6103 . . . . . . . . 9 (𝑗 = 𝐽 → (nei‘𝑗) = (nei‘𝐽))
65fveq1d 6105 . . . . . . . 8 (𝑗 = 𝐽 → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
76oveq1d 6564 . . . . . . 7 (𝑗 = 𝐽 → (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓) = (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))
87oveq2d 6565 . . . . . 6 (𝑗 = 𝐽 → (𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓)) = (𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)))
98fveq1d 6105 . . . . 5 (𝑗 = 𝐽 → ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓) = ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))
109xpeq2d 5063 . . . 4 (𝑗 = 𝐽 → ({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓)) = ({𝑥} × ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)))
114, 10iuneq12d 4482 . . 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)))
122, 11mpteq12dv 4663 . 2 (𝑗 = 𝐽 → (𝑓 ∈ ( 𝑘pm 𝑗) ↦ 𝑥 ∈ ((cls‘𝑗)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓))) = (𝑓 ∈ ( 𝑘pm 𝐽) ↦ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
13 unieq 4380 . . . 4 (𝑘 = 𝐾 𝑘 = 𝐾)
1413oveq1d 6564 . . 3 (𝑘 = 𝐾 → ( 𝑘pm 𝐽) = ( 𝐾pm 𝐽))
15 oveq1 6556 . . . . . 6 (𝑘 = 𝐾 → (𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)))
1615fveq1d 6105 . . . . 5 (𝑘 = 𝐾 → ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))
1716xpeq2d 5063 . . . 4 (𝑘 = 𝐾 → ({𝑥} × ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)) = ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)))
1817iuneq2d 4483 . . 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)))
1914, 18mpteq12dv 4663 . 2 (𝑘 = 𝐾 → (𝑓 ∈ ( 𝑘pm 𝐽) ↦ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))) = (𝑓 ∈ ( 𝐾pm 𝐽) ↦ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
20 df-cnext 21674 . 2 CnExt = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ ( 𝑘pm 𝑗) ↦ 𝑥 ∈ ((cls‘𝑗)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
21 ovex 6577 . . 3 ( 𝐾pm 𝐽) ∈ V
2221mptex 6390 . 2 (𝑓 ∈ ( 𝐾pm 𝐽) ↦ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))) ∈ V
2312, 19, 20, 22ovmpt2 6694 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽CnExt𝐾) = (𝑓 ∈ ( 𝐾pm 𝐽) ↦ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ‘cfv 5804  (class class class)co 6549   ↑pm cpm 7745   ↾t crest 15904  Topctop 20517  clsccl 20632  neicnei 20711   fLimf cflf 21549  CnExtccnext 21673 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-cnext 21674 This theorem is referenced by:  cnextfval  21676
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