Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  topfne Structured version   Visualization version   GIF version

Theorem topfne 34280
Description: Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.)
Hypotheses
Ref Expression
topfne.1 𝑋 = 𝐽
topfne.2 𝑌 = 𝐾
Assertion
Ref Expression
topfne ((𝐾 ∈ Top ∧ 𝑋 = 𝑌) → (𝐽𝐾𝐽Fne𝐾))

Proof of Theorem topfne
StepHypRef Expression
1 tgtop 21870 . . . 4 (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾)
21sseq2d 3933 . . 3 (𝐾 ∈ Top → (𝐽 ⊆ (topGen‘𝐾) ↔ 𝐽𝐾))
32bicomd 226 . 2 (𝐾 ∈ Top → (𝐽𝐾𝐽 ⊆ (topGen‘𝐾)))
4 topfne.1 . . . 4 𝑋 = 𝐽
5 topfne.2 . . . 4 𝑌 = 𝐾
64, 5isfne4 34266 . . 3 (𝐽Fne𝐾 ↔ (𝑋 = 𝑌𝐽 ⊆ (topGen‘𝐾)))
76baibr 540 . 2 (𝑋 = 𝑌 → (𝐽 ⊆ (topGen‘𝐾) ↔ 𝐽Fne𝐾))
83, 7sylan9bb 513 1 ((𝐾 ∈ Top ∧ 𝑋 = 𝑌) → (𝐽𝐾𝐽Fne𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wss 3866   cuni 4819   class class class wbr 5053  cfv 6380  topGenctg 16942  Topctop 21790  Fnecfne 34262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-topgen 16948  df-top 21791  df-fne 34263
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator