MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnextf Structured version   Visualization version   GIF version

Theorem cnextf 23570
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 23568 . . . 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 5888 . . . 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 4347 . . . . . . . 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 22835 . . . . . . . . . . . . . 14 (𝐾 ∈ Haus β†’ 𝐾 ∈ Top)
182, 17syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐾 ∈ Top)
195, 6cnextfval 23566 . . . . . . . . . . . . 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 5744 . . . . . . . . . . 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)))
3332eqabdv 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 6547 . . 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 5938 . . 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 6148 . . . 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 3479 . . . . . . . . 9 π‘₯ ∈ V
4039snnz 4781 . . . . . . . 8 {π‘₯} β‰  βˆ…
41 rnxp 6170 . . . . . . . 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 22419 . . . . . . . . . . 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 22419 . . . . . . . . . . . 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 23396 . . . . . . . . . . 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 23498 . . . . . . . . . . 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 3989 . . . . . . . . 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 4031 . . . . . 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 3147 . . . . 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 5049 . . . . 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 4031 . . 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 4021 . 2 (πœ‘ β†’ ran ((𝐽CnExt𝐾)β€˜πΉ) βŠ† 𝐡)
67 df-f 6548 . 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 2941  βˆ€wral 3062   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  βŸ¨cop 4635  βˆͺ cuni 4909  βˆͺ ciun 4998   Γ— cxp 5675  dom cdm 5677  ran crn 5678  Fun wfun 6538   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   β†Ύt crest 17366  Topctop 22395  TopOnctopon 22412  clsccl 22522  neicnei 22601  Hauscha 22812  Filcfil 23349   fLimf cflf 23439  CnExtccnext 23563
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-pm 8823  df-rest 17368  df-fbas 20941  df-fg 20942  df-top 22396  df-topon 22413  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-haus 22819  df-fil 23350  df-fm 23442  df-flim 23443  df-flf 23444  df-cnext 23564
This theorem is referenced by:  cnextcn  23571  cnextfres1  23572
  Copyright terms: Public domain W3C validator