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Theorem cnextval 21846
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 4435 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
21oveq2d 6651 . . 3 (𝑗 = 𝐽 → ( 𝑘pm 𝑗) = ( 𝑘pm 𝐽))
3 fveq2 6178 . . . . 5 (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
43fveq1d 6180 . . . 4 (𝑗 = 𝐽 → ((cls‘𝑗)‘dom 𝑓) = ((cls‘𝐽)‘dom 𝑓))
5 fveq2 6178 . . . . . . . . 9 (𝑗 = 𝐽 → (nei‘𝑗) = (nei‘𝐽))
65fveq1d 6180 . . . . . . . 8 (𝑗 = 𝐽 → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
76oveq1d 6650 . . . . . . 7 (𝑗 = 𝐽 → (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓) = (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))
87oveq2d 6651 . . . . . 6 (𝑗 = 𝐽 → (𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓)) = (𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)))
98fveq1d 6180 . . . . 5 (𝑗 = 𝐽 → ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓) = ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))
109xpeq2d 5129 . . . 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 4537 . . 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 4724 . 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 4435 . . . 4 (𝑘 = 𝐾 𝑘 = 𝐾)
1413oveq1d 6650 . . 3 (𝑘 = 𝐾 → ( 𝑘pm 𝐽) = ( 𝐾pm 𝐽))
15 oveq1 6642 . . . . . 6 (𝑘 = 𝐾 → (𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓)))
1615fveq1d 6180 . . . . 5 (𝑘 = 𝐾 → ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))
1716xpeq2d 5129 . . . 4 (𝑘 = 𝐾 → ({𝑥} × ((𝑘 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)) = ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓)))
1817iuneq2d 4538 . . 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 4724 . 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 21845 . 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 6663 . . 3 ( 𝐾pm 𝐽) ∈ V
2221mptex 6471 . 2 (𝑓 ∈ ( 𝐾pm 𝐽) ↦ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))) ∈ V
2312, 19, 20, 22ovmpt2 6781 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 384   = wceq 1481  wcel 1988  {csn 4168   cuni 4427   ciun 4511  cmpt 4720   × cxp 5102  dom cdm 5104  cfv 5876  (class class class)co 6635  pm cpm 7843  t crest 16062  Topctop 20679  clsccl 20803  neicnei 20882   fLimf cflf 21720  CnExtccnext 21844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-cnext 21845
This theorem is referenced by:  cnextfval  21847
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