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Theorem cnextf 23569
Description: Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
Hypotheses
Ref Expression
cnextf.1 𝐢 = βˆͺ 𝐽
cnextf.2 𝐡 = βˆͺ 𝐾
cnextf.3 (πœ‘ β†’ 𝐽 ∈ Top)
cnextf.4 (πœ‘ β†’ 𝐾 ∈ Haus)
cnextf.5 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
cnextf.a (πœ‘ β†’ 𝐴 βŠ† 𝐢)
cnextf.6 (πœ‘ β†’ ((clsβ€˜π½)β€˜π΄) = 𝐢)
cnextf.7 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) β‰  βˆ…)
Assertion
Ref Expression
cnextf (πœ‘ β†’ ((𝐽CnExt𝐾)β€˜πΉ):𝐢⟢𝐡)
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡   π‘₯,𝐢   π‘₯,𝐹   π‘₯,𝐽   π‘₯,𝐾   πœ‘,π‘₯

Proof of Theorem cnextf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cnextf.3 . . . 4 (πœ‘ β†’ 𝐽 ∈ Top)
2 cnextf.4 . . . 4 (πœ‘ β†’ 𝐾 ∈ Haus)
3 cnextf.5 . . . 4 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
4 cnextf.a . . . 4 (πœ‘ β†’ 𝐴 βŠ† 𝐢)
5 cnextf.1 . . . . 5 𝐢 = βˆͺ 𝐽
6 cnextf.2 . . . . 5 𝐡 = βˆͺ 𝐾
75, 6cnextfun 23567 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝐢)) β†’ Fun ((𝐽CnExt𝐾)β€˜πΉ))
81, 2, 3, 4, 7syl22anc 837 . . 3 (πœ‘ β†’ Fun ((𝐽CnExt𝐾)β€˜πΉ))
9 dfdm3 5887 . . . 4 dom ((𝐽CnExt𝐾)β€˜πΉ) = {π‘₯ ∣ βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ)}
10 simpl 483 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ πœ‘)
11 cnextf.6 . . . . . . . . 9 (πœ‘ β†’ ((clsβ€˜π½)β€˜π΄) = 𝐢)
1211eleq2d 2819 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜π΄) ↔ π‘₯ ∈ 𝐢))
1312biimpar 478 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄))
14 cnextf.7 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) β‰  βˆ…)
15 n0 4346 . . . . . . . 8 (((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) β‰  βˆ… ↔ βˆƒπ‘¦ 𝑦 ∈ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ))
1614, 15sylib 217 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ βˆƒπ‘¦ 𝑦 ∈ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ))
17 haustop 22834 . . . . . . . . . . . . . 14 (𝐾 ∈ Haus β†’ 𝐾 ∈ Top)
182, 17syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐾 ∈ Top)
195, 6cnextfval 23565 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝐢)) 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)))
201, 18, 3, 4, 19syl22anc 837 . . . . . . . . . . . 12 (πœ‘ 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)))
2120eleq2d 2819 . . . . . . . . . . 11 (πœ‘ β†’ (⟨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ) ↔ ⟨π‘₯, π‘¦βŸ© ∈ βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ))))
22 opeliunxp 5743 . . . . . . . . . . 11 (⟨π‘₯, π‘¦βŸ© ∈ βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) ↔ (π‘₯ ∈ ((clsβ€˜π½)β€˜π΄) ∧ 𝑦 ∈ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)))
2321, 22bitrdi 286 . . . . . . . . . 10 (πœ‘ β†’ (⟨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ) ↔ (π‘₯ ∈ ((clsβ€˜π½)β€˜π΄) ∧ 𝑦 ∈ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ))))
2423exbidv 1924 . . . . . . . . 9 (πœ‘ β†’ (βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ) ↔ βˆƒπ‘¦(π‘₯ ∈ ((clsβ€˜π½)β€˜π΄) ∧ 𝑦 ∈ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ))))
25 19.42v 1957 . . . . . . . . 9 (βˆƒπ‘¦(π‘₯ ∈ ((clsβ€˜π½)β€˜π΄) ∧ 𝑦 ∈ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) ↔ (π‘₯ ∈ ((clsβ€˜π½)β€˜π΄) ∧ βˆƒπ‘¦ 𝑦 ∈ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)))
2624, 25bitrdi 286 . . . . . . . 8 (πœ‘ β†’ (βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ) ↔ (π‘₯ ∈ ((clsβ€˜π½)β€˜π΄) ∧ βˆƒπ‘¦ 𝑦 ∈ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ))))
2726biimpar 478 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ((clsβ€˜π½)β€˜π΄) ∧ βˆƒπ‘¦ 𝑦 ∈ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ))) β†’ βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ))
2810, 13, 16, 27syl12anc 835 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ))
2926simprbda 499 . . . . . . 7 ((πœ‘ ∧ βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ)) β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄))
3012adantr 481 . . . . . . 7 ((πœ‘ ∧ βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ)) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜π΄) ↔ π‘₯ ∈ 𝐢))
3129, 30mpbid 231 . . . . . 6 ((πœ‘ ∧ βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ)) β†’ π‘₯ ∈ 𝐢)
3228, 31impbida 799 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ 𝐢 ↔ βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ)))
3332eqabdv 2867 . . . 4 (πœ‘ β†’ 𝐢 = {π‘₯ ∣ βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ)})
349, 33eqtr4id 2791 . . 3 (πœ‘ β†’ dom ((𝐽CnExt𝐾)β€˜πΉ) = 𝐢)
35 df-fn 6546 . . 3 (((𝐽CnExt𝐾)β€˜πΉ) Fn 𝐢 ↔ (Fun ((𝐽CnExt𝐾)β€˜πΉ) ∧ dom ((𝐽CnExt𝐾)β€˜πΉ) = 𝐢))
368, 34, 35sylanbrc 583 . 2 (πœ‘ β†’ ((𝐽CnExt𝐾)β€˜πΉ) Fn 𝐢)
3720rneqd 5937 . . 3 (πœ‘ β†’ ran ((𝐽CnExt𝐾)β€˜πΉ) = ran βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)))
38 rniun 6147 . . . 4 ran βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) = βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)ran ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ))
39 vex 3478 . . . . . . . . 9 π‘₯ ∈ V
4039snnz 4780 . . . . . . . 8 {π‘₯} β‰  βˆ…
41 rnxp 6169 . . . . . . . 8 ({π‘₯} β‰  βˆ… β†’ ran ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) = ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ))
4240, 41ax-mp 5 . . . . . . 7 ran ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) = ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)
4312biimpa 477 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)) β†’ π‘₯ ∈ 𝐢)
446toptopon 22418 . . . . . . . . . . 11 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜π΅))
4518, 44sylib 217 . . . . . . . . . 10 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π΅))
4645adantr 481 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐾 ∈ (TopOnβ€˜π΅))
475toptopon 22418 . . . . . . . . . . . 12 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜πΆ))
481, 47sylib 217 . . . . . . . . . . 11 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜πΆ))
4948adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐽 ∈ (TopOnβ€˜πΆ))
504adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐴 βŠ† 𝐢)
51 simpr 485 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ π‘₯ ∈ 𝐢)
52 trnei 23395 . . . . . . . . . . 11 ((𝐽 ∈ (TopOnβ€˜πΆ) ∧ 𝐴 βŠ† 𝐢 ∧ π‘₯ ∈ 𝐢) β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜π΄) ↔ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (Filβ€˜π΄)))
5352biimpa 477 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜πΆ) ∧ 𝐴 βŠ† 𝐢 ∧ π‘₯ ∈ 𝐢) ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)) β†’ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (Filβ€˜π΄))
5449, 50, 51, 13, 53syl31anc 1373 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (Filβ€˜π΄))
553adantr 481 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐹:𝐴⟢𝐡)
56 flfelbas 23497 . . . . . . . . . . 11 (((𝐾 ∈ (TopOnβ€˜π΅) ∧ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (Filβ€˜π΄) ∧ 𝐹:𝐴⟢𝐡) ∧ 𝑦 ∈ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) β†’ 𝑦 ∈ 𝐡)
5756ex 413 . . . . . . . . . 10 ((𝐾 ∈ (TopOnβ€˜π΅) ∧ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (Filβ€˜π΄) ∧ 𝐹:𝐴⟢𝐡) β†’ (𝑦 ∈ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) β†’ 𝑦 ∈ 𝐡))
5857ssrdv 3988 . . . . . . . . 9 ((𝐾 ∈ (TopOnβ€˜π΅) ∧ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (Filβ€˜π΄) ∧ 𝐹:𝐴⟢𝐡) β†’ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) βŠ† 𝐡)
5946, 54, 55, 58syl3anc 1371 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) βŠ† 𝐡)
6043, 59syldan 591 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)) β†’ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) βŠ† 𝐡)
6142, 60eqsstrid 4030 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)) β†’ ran ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) βŠ† 𝐡)
6261ralrimiva 3146 . . . . 5 (πœ‘ β†’ βˆ€π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)ran ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) βŠ† 𝐡)
63 iunss 5048 . . . . 5 (βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)ran ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) βŠ† 𝐡 ↔ βˆ€π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)ran ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) βŠ† 𝐡)
6462, 63sylibr 233 . . . 4 (πœ‘ β†’ βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)ran ({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) βŠ† 𝐡)
6538, 64eqsstrid 4030 . . 3 (πœ‘ β†’ ran βˆͺ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)({π‘₯} Γ— ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ)) βŠ† 𝐡)
6637, 65eqsstrd 4020 . 2 (πœ‘ β†’ ran ((𝐽CnExt𝐾)β€˜πΉ) βŠ† 𝐡)
67 df-f 6547 . 2 (((𝐽CnExt𝐾)β€˜πΉ):𝐢⟢𝐡 ↔ (((𝐽CnExt𝐾)β€˜πΉ) Fn 𝐢 ∧ ran ((𝐽CnExt𝐾)β€˜πΉ) βŠ† 𝐡))
6836, 66, 67sylanbrc 583 1 (πœ‘ β†’ ((𝐽CnExt𝐾)β€˜πΉ):𝐢⟢𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {cab 2709   β‰  wne 2940  βˆ€wral 3061   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  βŸ¨cop 4634  βˆͺ cuni 4908  βˆͺ ciun 4997   Γ— cxp 5674  dom cdm 5676  ran crn 5677  Fun wfun 6537   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   β†Ύt crest 17365  Topctop 22394  TopOnctopon 22411  clsccl 22521  neicnei 22600  Hauscha 22811  Filcfil 23348   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-nel 3047  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-int 4951  df-iun 4999  df-iin 5000  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-1st 7974  df-2nd 7975  df-map 8821  df-pm 8822  df-rest 17367  df-fbas 20940  df-fg 20941  df-top 22395  df-topon 22412  df-cld 22522  df-ntr 22523  df-cls 22524  df-nei 22601  df-haus 22818  df-fil 23349  df-fm 23441  df-flim 23442  df-flf 23443  df-cnext 23563
This theorem is referenced by:  cnextcn  23570  cnextfres1  23571
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