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Theorem cnextfval 23984
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 23983 . . 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 479 . 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 483 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ 𝑓 = 𝐹)
43dmeqd 5902 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ dom 𝑓 = dom 𝐹)
5 simplrl 775 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ 𝐹:𝐴⟢𝐡)
65fdmd 6728 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ dom 𝐹 = 𝐴)
74, 6eqtrd 2765 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ dom 𝑓 = 𝐴)
87fveq2d 6896 . . 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 7432 . . . . . 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 7432 . . . . 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 6899 . . . 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 5702 . . 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 5019 . 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 7743 . . . 4 (𝐾 ∈ Top β†’ βˆͺ 𝐾 ∈ V)
1514ad2antlr 725 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ βˆͺ 𝐾 ∈ V)
16 uniexg 7743 . . . 4 (𝐽 ∈ Top β†’ βˆͺ 𝐽 ∈ V)
1716ad2antrr 724 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ βˆͺ 𝐽 ∈ V)
18 eqid 2725 . . . . . 6 𝐴 = 𝐴
19 cnextfval.2 . . . . . 6 𝐡 = βˆͺ 𝐾
2018, 19feq23i 6711 . . . . 5 (𝐹:𝐴⟢𝐡 ↔ 𝐹:𝐴⟢βˆͺ 𝐾)
2120biimpi 215 . . . 4 (𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴⟢βˆͺ 𝐾)
2221ad2antrl 726 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐹:𝐴⟢βˆͺ 𝐾)
23 cnextfval.1 . . . . . 6 𝑋 = βˆͺ 𝐽
2423sseq2i 4002 . . . . 5 (𝐴 βŠ† 𝑋 ↔ 𝐴 βŠ† βˆͺ 𝐽)
2524biimpi 215 . . . 4 (𝐴 βŠ† 𝑋 β†’ 𝐴 βŠ† βˆͺ 𝐽)
2625ad2antll 727 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐴 βŠ† βˆͺ 𝐽)
27 elpm2r 8862 . . 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 837 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐹 ∈ (βˆͺ 𝐾 ↑pm βˆͺ 𝐽))
29 fvex 6905 . . . 4 ((clsβ€˜π½)β€˜π΄) ∈ V
30 vsnex 5425 . . . . 5 {π‘₯} ∈ V
31 fvex 6905 . . . . 5 ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) ∈ V
3230, 31xpex 7753 . . . 4 ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) ∈ V
3329, 32iunex 7970 . . 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 7007 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 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   βŠ† wss 3939  {csn 4624  βˆͺ cuni 4903  βˆͺ ciun 4991   ↦ cmpt 5226   Γ— cxp 5670  dom cdm 5672  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7416   ↑pm cpm 8844   β†Ύt crest 17401  Topctop 22813  clsccl 22940  neicnei 23019   fLimf cflf 23857  CnExtccnext 23981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7419  df-oprab 7420  df-mpo 7421  df-pm 8846  df-cnext 23982
This theorem is referenced by:  cnextrel  23985  cnextfun  23986  cnextfvval  23987  cnextf  23988  cnextfres  23991
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