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Theorem cncls 23000
Description: Continuity in terms of closure. (Contributed by Jeff Hankins, 1-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
cncls ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝒫 𝑋(𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯)))))
Distinct variable groups:   π‘₯,𝐹   π‘₯,𝐽   π‘₯,𝐾   π‘₯,𝑋   π‘₯,π‘Œ

Proof of Theorem cncls
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cnf2 22975 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
213expia 1119 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:π‘‹βŸΆπ‘Œ))
3 elpwi 4610 . . . . . . 7 (π‘₯ ∈ 𝒫 𝑋 β†’ π‘₯ βŠ† 𝑋)
43adantl 480 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ π‘₯ ∈ 𝒫 𝑋) β†’ π‘₯ βŠ† 𝑋)
5 toponuni 22638 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
65ad2antrr 722 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ π‘₯ ∈ 𝒫 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
74, 6sseqtrd 4023 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ π‘₯ ∈ 𝒫 𝑋) β†’ π‘₯ βŠ† βˆͺ 𝐽)
8 eqid 2730 . . . . . . 7 βˆͺ 𝐽 = βˆͺ 𝐽
98cnclsi 22998 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ π‘₯ βŠ† βˆͺ 𝐽) β†’ (𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯)))
109expcom 412 . . . . 5 (π‘₯ βŠ† βˆͺ 𝐽 β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ (𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯))))
117, 10syl 17 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ π‘₯ ∈ 𝒫 𝑋) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ (𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯))))
1211ralrimdva 3152 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ βˆ€π‘₯ ∈ 𝒫 𝑋(𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯))))
132, 12jcad 511 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝒫 𝑋(𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯)))))
14 toponmax 22650 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
1514ad3antrrr 726 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ 𝑋 ∈ 𝐽)
16 cnvimass 6081 . . . . . . . . 9 (◑𝐹 β€œ 𝑦) βŠ† dom 𝐹
17 fdm 6727 . . . . . . . . . 10 (𝐹:π‘‹βŸΆπ‘Œ β†’ dom 𝐹 = 𝑋)
1817ad2antlr 723 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ dom 𝐹 = 𝑋)
1916, 18sseqtrid 4035 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ (◑𝐹 β€œ 𝑦) βŠ† 𝑋)
2015, 19sselpwd 5327 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ (◑𝐹 β€œ 𝑦) ∈ 𝒫 𝑋)
21 fveq2 6892 . . . . . . . . . 10 (π‘₯ = (◑𝐹 β€œ 𝑦) β†’ ((clsβ€˜π½)β€˜π‘₯) = ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦)))
2221imaeq2d 6060 . . . . . . . . 9 (π‘₯ = (◑𝐹 β€œ 𝑦) β†’ (𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) = (𝐹 β€œ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦))))
23 imaeq2 6056 . . . . . . . . . 10 (π‘₯ = (◑𝐹 β€œ 𝑦) β†’ (𝐹 β€œ π‘₯) = (𝐹 β€œ (◑𝐹 β€œ 𝑦)))
2423fveq2d 6896 . . . . . . . . 9 (π‘₯ = (◑𝐹 β€œ 𝑦) β†’ ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯)) = ((clsβ€˜πΎ)β€˜(𝐹 β€œ (◑𝐹 β€œ 𝑦))))
2522, 24sseq12d 4016 . . . . . . . 8 (π‘₯ = (◑𝐹 β€œ 𝑦) β†’ ((𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯)) ↔ (𝐹 β€œ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦))) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ (◑𝐹 β€œ 𝑦)))))
2625rspcv 3609 . . . . . . 7 ((◑𝐹 β€œ 𝑦) ∈ 𝒫 𝑋 β†’ (βˆ€π‘₯ ∈ 𝒫 𝑋(𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯)) β†’ (𝐹 β€œ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦))) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ (◑𝐹 β€œ 𝑦)))))
2720, 26syl 17 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ (βˆ€π‘₯ ∈ 𝒫 𝑋(𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯)) β†’ (𝐹 β€œ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦))) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ (◑𝐹 β€œ 𝑦)))))
28 topontop 22637 . . . . . . . . . 10 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
2928ad3antlr 727 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ 𝐾 ∈ Top)
30 elpwi 4610 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 π‘Œ β†’ 𝑦 βŠ† π‘Œ)
3130adantl 480 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ 𝑦 βŠ† π‘Œ)
32 toponuni 22638 . . . . . . . . . . 11 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
3332ad3antlr 727 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
3431, 33sseqtrd 4023 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ 𝑦 βŠ† βˆͺ 𝐾)
35 ffun 6721 . . . . . . . . . . . 12 (𝐹:π‘‹βŸΆπ‘Œ β†’ Fun 𝐹)
3635ad2antlr 723 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ Fun 𝐹)
37 funimacnv 6630 . . . . . . . . . . 11 (Fun 𝐹 β†’ (𝐹 β€œ (◑𝐹 β€œ 𝑦)) = (𝑦 ∩ ran 𝐹))
3836, 37syl 17 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ (𝐹 β€œ (◑𝐹 β€œ 𝑦)) = (𝑦 ∩ ran 𝐹))
39 inss1 4229 . . . . . . . . . 10 (𝑦 ∩ ran 𝐹) βŠ† 𝑦
4038, 39eqsstrdi 4037 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ (𝐹 β€œ (◑𝐹 β€œ 𝑦)) βŠ† 𝑦)
41 eqid 2730 . . . . . . . . . 10 βˆͺ 𝐾 = βˆͺ 𝐾
4241clsss 22780 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑦 βŠ† βˆͺ 𝐾 ∧ (𝐹 β€œ (◑𝐹 β€œ 𝑦)) βŠ† 𝑦) β†’ ((clsβ€˜πΎ)β€˜(𝐹 β€œ (◑𝐹 β€œ 𝑦))) βŠ† ((clsβ€˜πΎ)β€˜π‘¦))
4329, 34, 40, 42syl3anc 1369 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ ((clsβ€˜πΎ)β€˜(𝐹 β€œ (◑𝐹 β€œ 𝑦))) βŠ† ((clsβ€˜πΎ)β€˜π‘¦))
44 sstr2 3990 . . . . . . . 8 ((𝐹 β€œ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦))) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ (◑𝐹 β€œ 𝑦))) β†’ (((clsβ€˜πΎ)β€˜(𝐹 β€œ (◑𝐹 β€œ 𝑦))) βŠ† ((clsβ€˜πΎ)β€˜π‘¦) β†’ (𝐹 β€œ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦))) βŠ† ((clsβ€˜πΎ)β€˜π‘¦)))
4543, 44syl5com 31 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ ((𝐹 β€œ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦))) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ (◑𝐹 β€œ 𝑦))) β†’ (𝐹 β€œ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦))) βŠ† ((clsβ€˜πΎ)β€˜π‘¦)))
46 topontop 22637 . . . . . . . . . . 11 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
4746ad3antrrr 726 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ 𝐽 ∈ Top)
485ad3antrrr 726 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ 𝑋 = βˆͺ 𝐽)
4918, 48eqtrd 2770 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ dom 𝐹 = βˆͺ 𝐽)
5016, 49sseqtrid 4035 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ (◑𝐹 β€œ 𝑦) βŠ† βˆͺ 𝐽)
518clsss3 22785 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (◑𝐹 β€œ 𝑦) βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦)) βŠ† βˆͺ 𝐽)
5247, 50, 51syl2anc 582 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦)) βŠ† βˆͺ 𝐽)
5352, 49sseqtrrd 4024 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦)) βŠ† dom 𝐹)
54 funimass3 7056 . . . . . . . 8 ((Fun 𝐹 ∧ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦)) βŠ† dom 𝐹) β†’ ((𝐹 β€œ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦))) βŠ† ((clsβ€˜πΎ)β€˜π‘¦) ↔ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘¦))))
5536, 53, 54syl2anc 582 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ ((𝐹 β€œ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦))) βŠ† ((clsβ€˜πΎ)β€˜π‘¦) ↔ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘¦))))
5645, 55sylibd 238 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ ((𝐹 β€œ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦))) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ (◑𝐹 β€œ 𝑦))) β†’ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘¦))))
5727, 56syld 47 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ 𝑦 ∈ 𝒫 π‘Œ) β†’ (βˆ€π‘₯ ∈ 𝒫 𝑋(𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯)) β†’ ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘¦))))
5857ralrimdva 3152 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘₯ ∈ 𝒫 𝑋(𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯)) β†’ βˆ€π‘¦ ∈ 𝒫 π‘Œ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘¦))))
5958imdistanda 570 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝒫 𝑋(𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯))) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ 𝒫 π‘Œ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘¦)))))
60 cncls2 22999 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ 𝒫 π‘Œ((clsβ€˜π½)β€˜(◑𝐹 β€œ 𝑦)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘¦)))))
6159, 60sylibrd 258 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝒫 𝑋(𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯))) β†’ 𝐹 ∈ (𝐽 Cn 𝐾)))
6213, 61impbid 211 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝒫 𝑋(𝐹 β€œ ((clsβ€˜π½)β€˜π‘₯)) βŠ† ((clsβ€˜πΎ)β€˜(𝐹 β€œ π‘₯)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β€œ cima 5680  Fun wfun 6538  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7413  Topctop 22617  TopOnctopon 22634  clsccl 22744   Cn ccn 22950
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
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 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 7416  df-oprab 7417  df-mpo 7418  df-map 8826  df-top 22618  df-topon 22635  df-cld 22745  df-cls 22747  df-cn 22953
This theorem is referenced by: (None)
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