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Theorem cncls2 22777
Description: Continuity in terms of closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
cncls2 ((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 π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯)))))
Distinct variable groups:   π‘₯,𝐹   π‘₯,𝐽   π‘₯,𝐾   π‘₯,𝑋   π‘₯,π‘Œ
Proof of Theorem cncls2
StepHypRef Expression
1 cnf2 22753 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
213expia 1122 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:π‘‹βŸΆπ‘Œ))
3 elpwi 4610 . . . . . . 7 (π‘₯ ∈ 𝒫 π‘Œ β†’ π‘₯ βŠ† π‘Œ)
43adantl 483 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ π‘₯ ∈ 𝒫 π‘Œ) β†’ π‘₯ βŠ† π‘Œ)
5 toponuni 22416 . . . . . . 7 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
65ad2antlr 726 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ π‘₯ ∈ 𝒫 π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
74, 6sseqtrd 4023 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ π‘₯ ∈ 𝒫 π‘Œ) β†’ π‘₯ βŠ† βˆͺ 𝐾)
8 eqid 2733 . . . . . . 7 βˆͺ 𝐾 = βˆͺ 𝐾
98cncls2i 22774 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ π‘₯ βŠ† βˆͺ 𝐾) β†’ ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯)))
109expcom 415 . . . . 5 (π‘₯ βŠ† βˆͺ 𝐾 β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯))))
117, 10syl 17 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ π‘₯ ∈ 𝒫 π‘Œ) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯))))
1211ralrimdva 3155 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ βˆ€π‘₯ ∈ 𝒫 π‘Œ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯))))
132, 12jcad 514 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝒫 π‘Œ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯)))))
148cldss2 22534 . . . . . . . . 9 (Clsdβ€˜πΎ) βŠ† 𝒫 βˆͺ 𝐾
155ad2antlr 726 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
1615pweqd 4620 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ 𝒫 π‘Œ = 𝒫 βˆͺ 𝐾)
1714, 16sseqtrrid 4036 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (Clsdβ€˜πΎ) βŠ† 𝒫 π‘Œ)
1817sseld 3982 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (π‘₯ ∈ (Clsdβ€˜πΎ) β†’ π‘₯ ∈ 𝒫 π‘Œ))
1918imim1d 82 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((π‘₯ ∈ 𝒫 π‘Œ β†’ ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯))) β†’ (π‘₯ ∈ (Clsdβ€˜πΎ) β†’ ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯)))))
20 cldcls 22546 . . . . . . . . . . . 12 (π‘₯ ∈ (Clsdβ€˜πΎ) β†’ ((clsβ€˜πΎ)β€˜π‘₯) = π‘₯)
2120ad2antll 728 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ π‘₯ ∈ (Clsdβ€˜πΎ))) β†’ ((clsβ€˜πΎ)β€˜π‘₯) = π‘₯)
2221imaeq2d 6060 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ π‘₯ ∈ (Clsdβ€˜πΎ))) β†’ (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯)) = (◑𝐹 β€œ π‘₯))
2322sseq2d 4015 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ π‘₯ ∈ (Clsdβ€˜πΎ))) β†’ (((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯)) ↔ ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ π‘₯)))
24 topontop 22415 . . . . . . . . . . 11 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2524ad2antrr 725 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ π‘₯ ∈ (Clsdβ€˜πΎ))) β†’ 𝐽 ∈ Top)
26 cnvimass 6081 . . . . . . . . . . 11 (◑𝐹 β€œ π‘₯) βŠ† dom 𝐹
27 fdm 6727 . . . . . . . . . . . . 13 (𝐹:π‘‹βŸΆπ‘Œ β†’ dom 𝐹 = 𝑋)
2827ad2antrl 727 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ π‘₯ ∈ (Clsdβ€˜πΎ))) β†’ dom 𝐹 = 𝑋)
29 toponuni 22416 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
3029ad2antrr 725 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ π‘₯ ∈ (Clsdβ€˜πΎ))) β†’ 𝑋 = βˆͺ 𝐽)
3128, 30eqtrd 2773 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ π‘₯ ∈ (Clsdβ€˜πΎ))) β†’ dom 𝐹 = βˆͺ 𝐽)
3226, 31sseqtrid 4035 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ π‘₯ ∈ (Clsdβ€˜πΎ))) β†’ (◑𝐹 β€œ π‘₯) βŠ† βˆͺ 𝐽)
33 eqid 2733 . . . . . . . . . . 11 βˆͺ 𝐽 = βˆͺ 𝐽
3433iscld4 22569 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (◑𝐹 β€œ π‘₯) βŠ† βˆͺ 𝐽) β†’ ((◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜π½) ↔ ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ π‘₯)))
3525, 32, 34syl2anc 585 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ π‘₯ ∈ (Clsdβ€˜πΎ))) β†’ ((◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜π½) ↔ ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ π‘₯)))
3623, 35bitr4d 282 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝐹:π‘‹βŸΆπ‘Œ ∧ π‘₯ ∈ (Clsdβ€˜πΎ))) β†’ (((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯)) ↔ (◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜π½)))
3736expr 458 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (π‘₯ ∈ (Clsdβ€˜πΎ) β†’ (((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯)) ↔ (◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜π½))))
3837pm5.74d 273 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((π‘₯ ∈ (Clsdβ€˜πΎ) β†’ ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯))) ↔ (π‘₯ ∈ (Clsdβ€˜πΎ) β†’ (◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜π½))))
3919, 38sylibd 238 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((π‘₯ ∈ 𝒫 π‘Œ β†’ ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯))) β†’ (π‘₯ ∈ (Clsdβ€˜πΎ) β†’ (◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜π½))))
4039ralimdv2 3164 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘₯ ∈ 𝒫 π‘Œ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯)) β†’ βˆ€π‘₯ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜π½)))
4140imdistanda 573 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝒫 π‘Œ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯))) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜π½))))
42 iscncl 22773 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ π‘₯) ∈ (Clsdβ€˜π½))))
4341, 42sylibrd 259 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝒫 π‘Œ((clsβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯))) β†’ 𝐹 ∈ (𝐽 Cn 𝐾)))
4413, 43impbid 211 1 ((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 π‘₯)) βŠ† (◑𝐹 β€œ ((clsβ€˜πΎ)β€˜π‘₯)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909  β—‘ccnv 5676  dom cdm 5677   β€œ cima 5680  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Topctop 22395  TopOnctopon 22412  Clsdccld 22520  clsccl 22522   Cn ccn 22728
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-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-map 8822  df-top 22396  df-topon 22413  df-cld 22523  df-cls 22525  df-cn 22731
This theorem is referenced by:  cncls  22778
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