MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnntr Structured version   Visualization version   GIF version
Theorem cnntr 23001
Description: Continuity in terms of interior. (Contributed by Jeff Hankins, 2-Oct-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
cnntr ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝒫 π‘Œ(◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)))))
Distinct variable groups:   π‘₯,𝐹   π‘₯,𝐽   π‘₯,𝐾   π‘₯,𝑋   π‘₯,π‘Œ
Proof of Theorem cnntr
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 π‘Œ β†’ π‘₯ βŠ† π‘Œ)
43adantl 480 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ π‘₯ ∈ 𝒫 π‘Œ) β†’ π‘₯ βŠ† π‘Œ)
5 toponuni 22638 . . . . . . 7 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ = βˆͺ 𝐾)
65ad2antlr 723 . . . . . 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 2730 . . . . . . 7 βˆͺ 𝐾 = βˆͺ 𝐾
98cnntri 22997 . . . . . 6 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ π‘₯ βŠ† βˆͺ 𝐾) β†’ (◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)))
109expcom 412 . . . . 5 (π‘₯ βŠ† βˆͺ 𝐾 β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ (◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯))))
117, 10syl 17 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ π‘₯ ∈ 𝒫 π‘Œ) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ (◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯))))
1211ralrimdva 3152 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ βˆ€π‘₯ ∈ 𝒫 π‘Œ(◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)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 π‘Œ(◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)))))
14 toponss 22651 . . . . . . . . . 10 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ π‘₯ βŠ† π‘Œ)
15 velpw 4608 . . . . . . . . . 10 (π‘₯ ∈ 𝒫 π‘Œ ↔ π‘₯ βŠ† π‘Œ)
1614, 15sylibr 233 . . . . . . . . 9 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ π‘₯ ∈ 𝒫 π‘Œ)
1716ex 411 . . . . . . . 8 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ (π‘₯ ∈ 𝐾 β†’ π‘₯ ∈ 𝒫 π‘Œ))
1817ad2antlr 723 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (π‘₯ ∈ 𝐾 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†’ (◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯))) β†’ (π‘₯ ∈ 𝐾 β†’ (◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)))))
20 topontop 22637 . . . . . . . . . . 11 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2120ad3antrrr 726 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐽 ∈ Top)
22 cnvimass 6081 . . . . . . . . . . 11 (◑𝐹 β€œ π‘₯) βŠ† dom 𝐹
23 fdm 6727 . . . . . . . . . . . . 13 (𝐹:π‘‹βŸΆπ‘Œ β†’ dom 𝐹 = 𝑋)
2423ad2antlr 723 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ dom 𝐹 = 𝑋)
25 toponuni 22638 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
2625ad3antrrr 726 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝑋 = βˆͺ 𝐽)
2724, 26eqtrd 2770 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ dom 𝐹 = βˆͺ 𝐽)
2822, 27sseqtrid 4035 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ π‘₯) βŠ† βˆͺ 𝐽)
29 eqid 2730 . . . . . . . . . . 11 βˆͺ 𝐽 = βˆͺ 𝐽
3029ntrss2 22783 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (◑𝐹 β€œ π‘₯) βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ π‘₯))
3121, 28, 30syl2anc 582 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ π‘₯))
32 eqss 3998 . . . . . . . . . 10 (((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) = (◑𝐹 β€œ π‘₯) ↔ (((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ π‘₯) ∧ (◑𝐹 β€œ π‘₯) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯))))
3332baib 534 . . . . . . . . 9 (((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) βŠ† (◑𝐹 β€œ π‘₯) β†’ (((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) = (◑𝐹 β€œ π‘₯) ↔ (◑𝐹 β€œ π‘₯) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯))))
3431, 33syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ (((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) = (◑𝐹 β€œ π‘₯) ↔ (◑𝐹 β€œ π‘₯) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯))))
3529isopn3 22792 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (◑𝐹 β€œ π‘₯) βŠ† βˆͺ 𝐽) β†’ ((◑𝐹 β€œ π‘₯) ∈ 𝐽 ↔ ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) = (◑𝐹 β€œ π‘₯)))
3621, 28, 35syl2anc 582 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ ((◑𝐹 β€œ π‘₯) ∈ 𝐽 ↔ ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) = (◑𝐹 β€œ π‘₯)))
37 topontop 22637 . . . . . . . . . . . 12 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ 𝐾 ∈ Top)
3837ad3antlr 727 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ 𝐾 ∈ Top)
39 isopn3i 22808 . . . . . . . . . . 11 ((𝐾 ∈ Top ∧ π‘₯ ∈ 𝐾) β†’ ((intβ€˜πΎ)β€˜π‘₯) = π‘₯)
4038, 39sylancom 586 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ ((intβ€˜πΎ)β€˜π‘₯) = π‘₯)
4140imaeq2d 6060 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) = (◑𝐹 β€œ π‘₯))
4241sseq1d 4014 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ ((◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) ↔ (◑𝐹 β€œ π‘₯) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯))))
4334, 36, 423bitr4rd 311 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) ∧ π‘₯ ∈ 𝐾) β†’ ((◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) ↔ (◑𝐹 β€œ π‘₯) ∈ 𝐽))
4443pm5.74da 800 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((π‘₯ ∈ 𝐾 β†’ (◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯))) ↔ (π‘₯ ∈ 𝐾 β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
4519, 44sylibd 238 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ ((π‘₯ ∈ 𝒫 π‘Œ β†’ (◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯))) β†’ (π‘₯ ∈ 𝐾 β†’ (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
4645ralimdv2 3161 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ 𝐹:π‘‹βŸΆπ‘Œ) β†’ (βˆ€π‘₯ ∈ 𝒫 π‘Œ(◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯)) β†’ βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ π‘₯) ∈ 𝐽))
4746imdistanda 570 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝒫 π‘Œ(◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯))) β†’ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
48 iscn 22961 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
4947, 48sylibrd 258 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ ((𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝒫 π‘Œ(◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ π‘₯))) β†’ 𝐹 ∈ (𝐽 Cn 𝐾)))
5013, 49impbid 211 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝒫 π‘Œ(◑𝐹 β€œ ((intβ€˜πΎ)β€˜π‘₯)) βŠ† ((intβ€˜π½)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   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 7413  Topctop 22617  TopOnctopon 22634  intcnt 22743   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-iun 5000  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-ntr 22746  df-cn 22953
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator