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Theorem cnextfval 23921
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 23920 . . 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 480 . 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 484 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ 𝑓 = 𝐹)
43dmeqd 5899 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ dom 𝑓 = dom 𝐹)
5 simplrl 774 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ 𝐹:𝐴⟢𝐡)
65fdmd 6722 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ dom 𝐹 = 𝐴)
74, 6eqtrd 2766 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ dom 𝑓 = 𝐴)
87fveq2d 6889 . . 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 7421 . . . . . 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 7421 . . . . 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 6892 . . . 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 5699 . . 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 5018 . 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 7727 . . . 4 (𝐾 ∈ Top β†’ βˆͺ 𝐾 ∈ V)
1514ad2antlr 724 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ βˆͺ 𝐾 ∈ V)
16 uniexg 7727 . . . 4 (𝐽 ∈ Top β†’ βˆͺ 𝐽 ∈ V)
1716ad2antrr 723 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ βˆͺ 𝐽 ∈ V)
18 eqid 2726 . . . . . 6 𝐴 = 𝐴
19 cnextfval.2 . . . . . 6 𝐡 = βˆͺ 𝐾
2018, 19feq23i 6705 . . . . 5 (𝐹:𝐴⟢𝐡 ↔ 𝐹:𝐴⟢βˆͺ 𝐾)
2120biimpi 215 . . . 4 (𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴⟢βˆͺ 𝐾)
2221ad2antrl 725 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐹:𝐴⟢βˆͺ 𝐾)
23 cnextfval.1 . . . . . 6 𝑋 = βˆͺ 𝐽
2423sseq2i 4006 . . . . 5 (𝐴 βŠ† 𝑋 ↔ 𝐴 βŠ† βˆͺ 𝐽)
2524biimpi 215 . . . 4 (𝐴 βŠ† 𝑋 β†’ 𝐴 βŠ† βˆͺ 𝐽)
2625ad2antll 726 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐴 βŠ† βˆͺ 𝐽)
27 elpm2r 8841 . . 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 836 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐹 ∈ (βˆͺ 𝐾 ↑pm βˆͺ 𝐽))
29 fvex 6898 . . . 4 ((clsβ€˜π½)β€˜π΄) ∈ V
30 vsnex 5422 . . . . 5 {π‘₯} ∈ V
31 fvex 6898 . . . . 5 ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) ∈ V
3230, 31xpex 7737 . . . 4 ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) ∈ V
3329, 32iunex 7954 . . 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 6999 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 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   βŠ† wss 3943  {csn 4623  βˆͺ cuni 4902  βˆͺ ciun 4990   ↦ cmpt 5224   Γ— cxp 5667  dom cdm 5669  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405   ↑pm cpm 8823   β†Ύt crest 17375  Topctop 22750  clsccl 22877  neicnei 22956   fLimf cflf 23794  CnExtccnext 23918
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-pm 8825  df-cnext 23919
This theorem is referenced by:  cnextrel  23922  cnextfun  23923  cnextfvval  23924  cnextf  23925  cnextfres  23928
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