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Theorem cnextfval 23429
Description: The continuous extension of a given function 𝐹. (Contributed by Thierry Arnoux, 1-Dec-2017.)
Hypotheses
Ref Expression
cnextfval.1 𝑋 = βˆͺ 𝐽
cnextfval.2 𝐡 = βˆͺ 𝐾
Assertion
Ref Expression
cnextfval (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ ((𝐽CnExt𝐾)β€˜πΉ) = βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)))
Distinct variable groups:   π‘₯,𝐽   π‘₯,𝐾   π‘₯,𝐴   π‘₯,𝐡   π‘₯,𝐹   π‘₯,𝑋

Proof of Theorem cnextfval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 cnextval 23428 . . 3 ((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‘“))))
21adantr 482 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) 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‘“))))
3 simpr 486 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ 𝑓 = 𝐹)
43dmeqd 5866 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ dom 𝑓 = dom 𝐹)
5 simplrl 776 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ 𝐹:𝐴⟢𝐡)
65fdmd 6684 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ dom 𝐹 = 𝐴)
74, 6eqtrd 2777 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ dom 𝑓 = 𝐴)
87fveq2d 6851 . . 3 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ ((clsβ€˜π½)β€˜dom 𝑓) = ((clsβ€˜π½)β€˜π΄))
97oveq2d 7378 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓) = (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))
109oveq2d 7378 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ (𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓)) = (𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴)))
1110, 3fveq12d 6854 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“) = ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ))
1211xpeq2d 5668 . . 3 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“)) = ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)))
138, 12iuneq12d 4987 . 2 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜dom 𝑓)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt dom 𝑓))β€˜π‘“)) = βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)))
14 uniexg 7682 . . . 4 (𝐾 ∈ Top β†’ βˆͺ 𝐾 ∈ V)
1514ad2antlr 726 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ βˆͺ 𝐾 ∈ V)
16 uniexg 7682 . . . 4 (𝐽 ∈ Top β†’ βˆͺ 𝐽 ∈ V)
1716ad2antrr 725 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ βˆͺ 𝐽 ∈ V)
18 eqid 2737 . . . . . 6 𝐴 = 𝐴
19 cnextfval.2 . . . . . 6 𝐡 = βˆͺ 𝐾
2018, 19feq23i 6667 . . . . 5 (𝐹:𝐴⟢𝐡 ↔ 𝐹:𝐴⟢βˆͺ 𝐾)
2120biimpi 215 . . . 4 (𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴⟢βˆͺ 𝐾)
2221ad2antrl 727 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐹:𝐴⟢βˆͺ 𝐾)
23 cnextfval.1 . . . . . 6 𝑋 = βˆͺ 𝐽
2423sseq2i 3978 . . . . 5 (𝐴 βŠ† 𝑋 ↔ 𝐴 βŠ† βˆͺ 𝐽)
2524biimpi 215 . . . 4 (𝐴 βŠ† 𝑋 β†’ 𝐴 βŠ† βˆͺ 𝐽)
2625ad2antll 728 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐴 βŠ† βˆͺ 𝐽)
27 elpm2r 8790 . . 3 (((βˆͺ 𝐾 ∈ V ∧ βˆͺ 𝐽 ∈ V) ∧ (𝐹:𝐴⟢βˆͺ 𝐾 ∧ 𝐴 βŠ† βˆͺ 𝐽)) β†’ 𝐹 ∈ (βˆͺ 𝐾 ↑pm βˆͺ 𝐽))
2815, 17, 22, 26, 27syl22anc 838 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐹 ∈ (βˆͺ 𝐾 ↑pm βˆͺ 𝐽))
29 fvex 6860 . . . 4 ((clsβ€˜π½)β€˜π΄) ∈ V
30 vsnex 5391 . . . . 5 {π‘₯} ∈ V
31 fvex 6860 . . . . 5 ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) ∈ V
3230, 31xpex 7692 . . . 4 ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) ∈ V
3329, 32iunex 7906 . . 3 βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) ∈ V
3433a1i 11 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) ∈ V)
352, 13, 28, 34fvmptd 6960 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ ((𝐽CnExt𝐾)β€˜πΉ) = βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3448   βŠ† wss 3915  {csn 4591  βˆͺ cuni 4870  βˆͺ ciun 4959   ↦ cmpt 5193   Γ— cxp 5636  dom cdm 5638  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362   ↑pm cpm 8773   β†Ύt crest 17309  Topctop 22258  clsccl 22385  neicnei 22464   fLimf cflf 23302  CnExtccnext 23426
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-pm 8775  df-cnext 23427
This theorem is referenced by:  cnextrel  23430  cnextfun  23431  cnextfvval  23432  cnextf  23433  cnextfres  23436
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