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Theorem cnextfval 23565
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 23564 . . 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 481 . 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 485 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ 𝑓 = 𝐹)
43dmeqd 5905 . . . . 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 2772 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ dom 𝑓 = 𝐴)
87fveq2d 6895 . . 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 7424 . . . . . 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 7424 . . . . 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 6898 . . . 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 5706 . . 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 5025 . 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 7729 . . . 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 7729 . . . 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 2732 . . . . . 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 4011 . . . . 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 8838 . . 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 6904 . . . 4 ((clsβ€˜π½)β€˜π΄) ∈ V
30 vsnex 5429 . . . . 5 {π‘₯} ∈ V
31 fvex 6904 . . . . 5 ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) ∈ V
3230, 31xpex 7739 . . . 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 7005 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 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3948  {csn 4628  βˆͺ cuni 4908  βˆͺ ciun 4997   ↦ cmpt 5231   Γ— cxp 5674  dom cdm 5676  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ↑pm cpm 8820   β†Ύt crest 17365  Topctop 22394  clsccl 22521  neicnei 22600   fLimf cflf 23438  CnExtccnext 23562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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-pw 4604  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 7411  df-oprab 7412  df-mpo 7413  df-pm 8822  df-cnext 23563
This theorem is referenced by:  cnextrel  23566  cnextfun  23567  cnextfvval  23568  cnextf  23569  cnextfres  23572
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