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Theorem cnextf 23440
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 23438 . . . 4 (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟢𝐡 ∧ 𝐴 βŠ† 𝐢)) β†’ Fun ((𝐽CnExt𝐾)β€˜πΉ))
81, 2, 3, 4, 7syl22anc 838 . . 3 (πœ‘ β†’ Fun ((𝐽CnExt𝐾)β€˜πΉ))
9 dfdm3 5847 . . . 4 dom ((𝐽CnExt𝐾)β€˜πΉ) = {π‘₯ ∣ βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ)}
10 simpl 484 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ πœ‘)
11 cnextf.6 . . . . . . . . 9 (πœ‘ β†’ ((clsβ€˜π½)β€˜π΄) = 𝐢)
1211eleq2d 2820 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ ((clsβ€˜π½)β€˜π΄) ↔ π‘₯ ∈ 𝐢))
1312biimpar 479 . . . . . . 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 4310 . . . . . . . 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 22705 . . . . . . . . . . . . . 14 (𝐾 ∈ Haus β†’ 𝐾 ∈ Top)
182, 17syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐾 ∈ Top)
195, 6cnextfval 23436 . . . . . . . . . . . . 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 838 . . . . . . . . . . . 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 2820 . . . . . . . . . . 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 5703 . . . . . . . . . . 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 287 . . . . . . . . . 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 1925 . . . . . . . . 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 1958 . . . . . . . . 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 287 . . . . . . . 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 479 . . . . . . 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 836 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ))
2926simprbda 500 . . . . . . 7 ((πœ‘ ∧ βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ)) β†’ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄))
3012adantr 482 . . . . . . 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 800 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ 𝐢 ↔ βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ)))
3332abbi2dv 2868 . . . 4 (πœ‘ β†’ 𝐢 = {π‘₯ ∣ βˆƒπ‘¦βŸ¨π‘₯, π‘¦βŸ© ∈ ((𝐽CnExt𝐾)β€˜πΉ)})
349, 33eqtr4id 2792 . . 3 (πœ‘ β†’ dom ((𝐽CnExt𝐾)β€˜πΉ) = 𝐢)
35 df-fn 6503 . . 3 (((𝐽CnExt𝐾)β€˜πΉ) Fn 𝐢 ↔ (Fun ((𝐽CnExt𝐾)β€˜πΉ) ∧ dom ((𝐽CnExt𝐾)β€˜πΉ) = 𝐢))
368, 34, 35sylanbrc 584 . 2 (πœ‘ β†’ ((𝐽CnExt𝐾)β€˜πΉ) Fn 𝐢)
3720rneqd 5897 . . 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 6104 . . . 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 3451 . . . . . . . . 9 π‘₯ ∈ V
4039snnz 4741 . . . . . . . 8 {π‘₯} β‰  βˆ…
41 rnxp 6126 . . . . . . . 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 478 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ ((clsβ€˜π½)β€˜π΄)) β†’ π‘₯ ∈ 𝐢)
446toptopon 22289 . . . . . . . . . . 11 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜π΅))
4518, 44sylib 217 . . . . . . . . . 10 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π΅))
4645adantr 482 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐾 ∈ (TopOnβ€˜π΅))
475toptopon 22289 . . . . . . . . . . . 12 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜πΆ))
481, 47sylib 217 . . . . . . . . . . 11 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜πΆ))
4948adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐽 ∈ (TopOnβ€˜πΆ))
504adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐴 βŠ† 𝐢)
51 simpr 486 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ π‘₯ ∈ 𝐢)
52 trnei 23266 . . . . . . . . . . 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 478 . . . . . . . . . 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 1374 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴) ∈ (Filβ€˜π΄))
553adantr 482 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ 𝐹:𝐴⟢𝐡)
56 flfelbas 23368 . . . . . . . . . . 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 414 . . . . . . . . . 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 3954 . . . . . . . . 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 1372 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝐢) β†’ ((𝐾 fLimf (((neiβ€˜π½)β€˜{π‘₯}) β†Ύt 𝐴))β€˜πΉ) βŠ† 𝐡)
6043, 59syldan 592 . . . . . . 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 3996 . . . . . 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 3140 . . . . 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 5009 . . . . 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 3996 . . 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 3986 . 2 (πœ‘ β†’ ran ((𝐽CnExt𝐾)β€˜πΉ) βŠ† 𝐡)
67 df-f 6504 . 2 (((𝐽CnExt𝐾)β€˜πΉ):𝐢⟢𝐡 ↔ (((𝐽CnExt𝐾)β€˜πΉ) Fn 𝐢 ∧ ran ((𝐽CnExt𝐾)β€˜πΉ) βŠ† 𝐡))
6836, 66, 67sylanbrc 584 1 (πœ‘ β†’ ((𝐽CnExt𝐾)β€˜πΉ):𝐢⟢𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710   β‰  wne 2940  βˆ€wral 3061   βŠ† wss 3914  βˆ…c0 4286  {csn 4590  βŸ¨cop 4596  βˆͺ cuni 4869  βˆͺ ciun 4958   Γ— cxp 5635  dom cdm 5637  ran crn 5638  Fun wfun 6494   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   β†Ύt crest 17310  Topctop 22265  TopOnctopon 22282  clsccl 22392  neicnei 22471  Hauscha 22682  Filcfil 23219   fLimf cflf 23309  CnExtccnext 23433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-map 8773  df-pm 8774  df-rest 17312  df-fbas 20816  df-fg 20817  df-top 22266  df-topon 22283  df-cld 22393  df-ntr 22394  df-cls 22395  df-nei 22472  df-haus 22689  df-fil 23220  df-fm 23312  df-flim 23313  df-flf 23314  df-cnext 23434
This theorem is referenced by:  cnextcn  23441  cnextfres1  23442
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