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Theorem topfne 32324
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 20758 . . . 4 (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾)
21sseq2d 3625 . . 3 (𝐾 ∈ Top → (𝐽 ⊆ (topGen‘𝐾) ↔ 𝐽𝐾))
32bicomd 213 . 2 (𝐾 ∈ Top → (𝐽𝐾𝐽 ⊆ (topGen‘𝐾)))
4 topfne.1 . . . 4 𝑋 = 𝐽
5 topfne.2 . . . 4 𝑌 = 𝐾
64, 5isfne4 32310 . . 3 (𝐽Fne𝐾 ↔ (𝑋 = 𝑌𝐽 ⊆ (topGen‘𝐾)))
76baibr 944 . 2 (𝑋 = 𝑌 → (𝐽 ⊆ (topGen‘𝐾) ↔ 𝐽Fne𝐾))
83, 7sylan9bb 735 1 ((𝐾 ∈ Top ∧ 𝑋 = 𝑌) → (𝐽𝐾𝐽Fne𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wss 3567   cuni 4427   class class class wbr 4644  cfv 5876  topGenctg 16079  Topctop 20679  Fnecfne 32306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-topgen 16085  df-top 20680  df-fne 32307
This theorem is referenced by: (None)
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