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Theorem cnextfval 23786
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 23785 . . 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 5904 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ dom 𝑓 = dom 𝐹)
5 simplrl 773 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ 𝐹:𝐴⟢𝐡)
65fdmd 6727 . . . . 5 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ dom 𝐹 = 𝐴)
74, 6eqtrd 2770 . . . 4 ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) ∧ 𝑓 = 𝐹) β†’ dom 𝑓 = 𝐴)
87fveq2d 6894 . . 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 7427 . . . . . 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 7427 . . . . 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 6897 . . . 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 5705 . . 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 5024 . 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 7732 . . . 4 (𝐾 ∈ Top β†’ βˆͺ 𝐾 ∈ V)
1514ad2antlr 723 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ βˆͺ 𝐾 ∈ V)
16 uniexg 7732 . . . 4 (𝐽 ∈ Top β†’ βˆͺ 𝐽 ∈ V)
1716ad2antrr 722 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ βˆͺ 𝐽 ∈ V)
18 eqid 2730 . . . . . 6 𝐴 = 𝐴
19 cnextfval.2 . . . . . 6 𝐡 = βˆͺ 𝐾
2018, 19feq23i 6710 . . . . 5 (𝐹:𝐴⟢𝐡 ↔ 𝐹:𝐴⟢βˆͺ 𝐾)
2120biimpi 215 . . . 4 (𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴⟢βˆͺ 𝐾)
2221ad2antrl 724 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐹:𝐴⟢βˆͺ 𝐾)
23 cnextfval.1 . . . . . 6 𝑋 = βˆͺ 𝐽
2423sseq2i 4010 . . . . 5 (𝐴 βŠ† 𝑋 ↔ 𝐴 βŠ† βˆͺ 𝐽)
2524biimpi 215 . . . 4 (𝐴 βŠ† 𝑋 β†’ 𝐴 βŠ† βˆͺ 𝐽)
2625ad2antll 725 . . 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 835 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐹 ∈ (βˆͺ 𝐾 ↑pm βˆͺ 𝐽))
29 fvex 6903 . . . 4 ((clsβ€˜π½)β€˜π΄) ∈ V
30 vsnex 5428 . . . . 5 {π‘₯} ∈ V
31 fvex 6903 . . . . 5 ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) ∈ V
3230, 31xpex 7742 . . . 4 ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) ∈ V
3329, 32iunex 7957 . . 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 7004 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 1539   ∈ wcel 2104  Vcvv 3472   βŠ† wss 3947  {csn 4627  βˆͺ cuni 4907  βˆͺ ciun 4996   ↦ cmpt 5230   Γ— cxp 5673  dom cdm 5675  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ↑pm cpm 8823   β†Ύt crest 17370  Topctop 22615  clsccl 22742  neicnei 22821   fLimf cflf 23659  CnExtccnext 23783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-pm 8825  df-cnext 23784
This theorem is referenced by:  cnextrel  23787  cnextfun  23788  cnextfvval  23789  cnextf  23790  cnextfres  23793
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