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Theorem cnextval 23786
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 4919 . . . 4 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
21oveq2d 7428 . . 3 (𝑗 = 𝐽 β†’ (βˆͺ π‘˜ ↑pm βˆͺ 𝑗) = (βˆͺ π‘˜ ↑pm βˆͺ 𝐽))
3 fveq2 6891 . . . . 5 (𝑗 = 𝐽 β†’ (clsβ€˜π‘—) = (clsβ€˜π½))
43fveq1d 6893 . . . 4 (𝑗 = 𝐽 β†’ ((clsβ€˜π‘—)β€˜dom 𝑓) = ((clsβ€˜π½)β€˜dom 𝑓))
5 fveq2 6891 . . . . . . . . 9 (𝑗 = 𝐽 β†’ (neiβ€˜π‘—) = (neiβ€˜π½))
65fveq1d 6893 . . . . . . . 8 (𝑗 = 𝐽 β†’ ((neiβ€˜π‘—)β€˜{π‘₯}) = ((neiβ€˜π½)β€˜{π‘₯}))
76oveq1d 7427 . . . . . . 7 (𝑗 = 𝐽 β†’ (((neiβ€˜π‘—)β€˜{π‘₯}) β†Ύt dom 𝑓) = (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))
87oveq2d 7428 . . . . . 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 6893 . . . . 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 5706 . . . 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 5025 . . 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 5239 . 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 4919 . . . 4 (π‘˜ = 𝐾 β†’ βˆͺ π‘˜ = βˆͺ 𝐾)
1413oveq1d 7427 . . 3 (π‘˜ = 𝐾 β†’ (βˆͺ π‘˜ ↑pm βˆͺ 𝐽) = (βˆͺ 𝐾 ↑pm βˆͺ 𝐽))
15 oveq1 7419 . . . . . 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 6893 . . . . 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 5706 . . . 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 5026 . . 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 5239 . 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 23785 . 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 7445 . . 3 (βˆͺ 𝐾 ↑pm βˆͺ 𝐽) ∈ V
2221mptex 7227 . 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 7571 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 395   = wceq 1540   ∈ wcel 2105  {csn 4628  βˆͺ cuni 4908  βˆͺ ciun 4997   ↦ cmpt 5231   Γ— cxp 5674  dom cdm 5676  β€˜cfv 6543  (class class class)co 7412   ↑pm cpm 8824   β†Ύt crest 17371  Topctop 22616  clsccl 22743  neicnei 22822   fLimf cflf 23660  CnExtccnext 23784
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-cnext 23785
This theorem is referenced by:  cnextfval  23787
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