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Theorem cnextval 23785
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) β†’ (𝐽CnExt𝐾) = (𝑓 ∈ (βˆͺ 𝐾 ↑pm βˆͺ 𝐽) ↦ βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜dom 𝑓)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“))))
Distinct variable groups:   π‘₯,𝑓,𝐽   𝑓,𝐾,π‘₯
Proof of Theorem cnextval
Dummy variables 𝑗 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unieq 4918 . . . 4 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
21oveq2d 7427 . . 3 (𝑗 = 𝐽 β†’ (βˆͺ π‘˜ ↑pm βˆͺ 𝑗) = (βˆͺ π‘˜ ↑pm βˆͺ 𝐽))
3 fveq2 6890 . . . . 5 (𝑗 = 𝐽 β†’ (clsβ€˜π‘—) = (clsβ€˜π½))
43fveq1d 6892 . . . 4 (𝑗 = 𝐽 β†’ ((clsβ€˜π‘—)β€˜dom 𝑓) = ((clsβ€˜π½)β€˜dom 𝑓))
5 fveq2 6890 . . . . . . . . 9 (𝑗 = 𝐽 β†’ (neiβ€˜π‘—) = (neiβ€˜π½))
65fveq1d 6892 . . . . . . . 8 (𝑗 = 𝐽 β†’ ((neiβ€˜π‘—)β€˜{π‘₯}) = ((neiβ€˜π½)β€˜{π‘₯}))
76oveq1d 7426 . . . . . . 7 (𝑗 = 𝐽 β†’ (((neiβ€˜π‘—)β€˜{π‘₯}) β†Ύt dom 𝑓) = (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))
87oveq2d 7427 . . . . . 6 (𝑗 = 𝐽 β†’ (π‘˜ fLimf (((neiβ€˜π‘—)β€˜{π‘₯}) β†Ύt dom 𝑓)) = (π‘˜ fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓)))
98fveq1d 6892 . . . . 5 (𝑗 = 𝐽 β†’ ((π‘˜ fLimf (((neiβ€˜π‘—)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“) = ((π‘˜ fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“))
109xpeq2d 5705 . . . 4 (𝑗 = 𝐽 β†’ ({π‘₯} Γ— ((π‘˜ fLimf (((neiβ€˜π‘—)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“)) = ({π‘₯} Γ— ((π‘˜ fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“)))
114, 10iuneq12d 5024 . . 3 (𝑗 = 𝐽 β†’ βˆͺ π‘₯ ∈ ((clsβ€˜π‘—)β€˜dom 𝑓)({π‘₯} Γ— ((π‘˜ fLimf (((neiβ€˜π‘—)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“)) = βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜dom 𝑓)({π‘₯} Γ— ((π‘˜ fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“)))
122, 11mpteq12dv 5238 . 2 (𝑗 = 𝐽 β†’ (𝑓 ∈ (βˆͺ π‘˜ ↑pm βˆͺ 𝑗) ↦ βˆͺ π‘₯ ∈ ((clsβ€˜π‘—)β€˜dom 𝑓)({π‘₯} Γ— ((π‘˜ fLimf (((neiβ€˜π‘—)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“))) = (𝑓 ∈ (βˆͺ π‘˜ ↑pm βˆͺ 𝐽) ↦ βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜dom 𝑓)({π‘₯} Γ— ((π‘˜ fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“))))
13 unieq 4918 . . . 4 (π‘˜ = 𝐾 β†’ βˆͺ π‘˜ = βˆͺ 𝐾)
1413oveq1d 7426 . . 3 (π‘˜ = 𝐾 β†’ (βˆͺ π‘˜ ↑pm βˆͺ 𝐽) = (βˆͺ 𝐾 ↑pm βˆͺ 𝐽))
15 oveq1 7418 . . . . . 6 (π‘˜ = 𝐾 β†’ (π‘˜ fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓)) = (𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓)))
1615fveq1d 6892 . . . . 5 (π‘˜ = 𝐾 β†’ ((π‘˜ fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“) = ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“))
1716xpeq2d 5705 . . . 4 (π‘˜ = 𝐾 β†’ ({π‘₯} Γ— ((π‘˜ fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“)) = ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“)))
1817iuneq2d 5025 . . 3 (π‘˜ = 𝐾 β†’ βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜dom 𝑓)({π‘₯} Γ— ((π‘˜ fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“)) = βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜dom 𝑓)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“)))
1914, 18mpteq12dv 5238 . 2 (π‘˜ = 𝐾 β†’ (𝑓 ∈ (βˆͺ π‘˜ ↑pm βˆͺ 𝐽) ↦ βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜dom 𝑓)({π‘₯} Γ— ((π‘˜ fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“))) = (𝑓 ∈ (βˆͺ 𝐾 ↑pm βˆͺ 𝐽) ↦ βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜dom 𝑓)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“))))
20 df-cnext 23784 . 2 CnExt = (𝑗 ∈ Top, π‘˜ ∈ Top ↦ (𝑓 ∈ (βˆͺ π‘˜ ↑pm βˆͺ 𝑗) ↦ βˆͺ π‘₯ ∈ ((clsβ€˜π‘—)β€˜dom 𝑓)({π‘₯} Γ— ((π‘˜ fLimf (((neiβ€˜π‘—)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“))))
21 ovex 7444 . . 3 (βˆͺ 𝐾 ↑pm βˆͺ 𝐽) ∈ V
2221mptex 7226 . 2 (𝑓 ∈ (βˆͺ 𝐾 ↑pm βˆͺ 𝐽) ↦ βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜dom 𝑓)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“))) ∈ V
2312, 19, 20, 22ovmpo 7570 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) β†’ (𝐽CnExt𝐾) = (𝑓 ∈ (βˆͺ 𝐾 ↑pm βˆͺ 𝐽) ↦ βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜dom 𝑓)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {csn 4627  βˆͺ cuni 4907  βˆͺ ciun 4996   ↦ cmpt 5230   Γ— cxp 5673  dom cdm 5675  β€˜cfv 6542  (class class class)co 7411   ↑pm cpm 8823   β†Ύt crest 17370  Topctop 22615  clsccl 22742  neicnei 22821   fLimf cflf 23659  CnExtccnext 23783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-cnext 23784
This theorem is referenced by:  cnextfval  23786
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