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Theorem cnextval 23435
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 4880 . . . 4 (𝑗 = 𝐽 β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
21oveq2d 7377 . . 3 (𝑗 = 𝐽 β†’ (βˆͺ π‘˜ ↑pm βˆͺ 𝑗) = (βˆͺ π‘˜ ↑pm βˆͺ 𝐽))
3 fveq2 6846 . . . . 5 (𝑗 = 𝐽 β†’ (clsβ€˜π‘—) = (clsβ€˜π½))
43fveq1d 6848 . . . 4 (𝑗 = 𝐽 β†’ ((clsβ€˜π‘—)β€˜dom 𝑓) = ((clsβ€˜π½)β€˜dom 𝑓))
5 fveq2 6846 . . . . . . . . 9 (𝑗 = 𝐽 β†’ (neiβ€˜π‘—) = (neiβ€˜π½))
65fveq1d 6848 . . . . . . . 8 (𝑗 = 𝐽 β†’ ((neiβ€˜π‘—)β€˜{π‘₯}) = ((neiβ€˜π½)β€˜{π‘₯}))
76oveq1d 7376 . . . . . . 7 (𝑗 = 𝐽 β†’ (((neiβ€˜π‘—)β€˜{π‘₯}) β†Ύt dom 𝑓) = (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))
87oveq2d 7377 . . . . . 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 6848 . . . . 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 5667 . . . 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 4986 . . 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 5200 . 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 4880 . . . 4 (π‘˜ = 𝐾 β†’ βˆͺ π‘˜ = βˆͺ 𝐾)
1413oveq1d 7376 . . 3 (π‘˜ = 𝐾 β†’ (βˆͺ π‘˜ ↑pm βˆͺ 𝐽) = (βˆͺ 𝐾 ↑pm βˆͺ 𝐽))
15 oveq1 7368 . . . . . 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 6848 . . . . 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 5667 . . . 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 4987 . . 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 5200 . 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 23434 . 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 7394 . . 3 (βˆͺ 𝐾 ↑pm βˆͺ 𝐽) ∈ V
2221mptex 7177 . 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 7519 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 397   = wceq 1542   ∈ wcel 2107  {csn 4590  βˆͺ cuni 4869  βˆͺ ciun 4958   ↦ cmpt 5192   Γ— cxp 5635  dom cdm 5637  β€˜cfv 6500  (class class class)co 7361   ↑pm cpm 8772   β†Ύt crest 17310  Topctop 22265  clsccl 22392  neicnei 22471   fLimf cflf 23309  CnExtccnext 23433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-cnext 23434
This theorem is referenced by:  cnextfval  23436
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