Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  topfneec2 Structured version   Visualization version   GIF version

Theorem topfneec2 33761
Description: A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.)
Hypothesis
Ref Expression
topfneec2.1 = (Fne ∩ Fne)
Assertion
Ref Expression
topfneec2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ([𝐽] = [𝐾] 𝐽 = 𝐾))

Proof of Theorem topfneec2
StepHypRef Expression
1 topfneec2.1 . . 3 = (Fne ∩ Fne)
21fneval 33757 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 𝐾 ↔ (topGen‘𝐽) = (topGen‘𝐾)))
31fneer 33758 . . . 4 Er V
43a1i 11 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → Er V)
5 elex 3498 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
65adantr 484 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → 𝐽 ∈ V)
74, 6erth 8334 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 𝐾 ↔ [𝐽] = [𝐾] ))
8 tgtop 21581 . . 3 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
9 tgtop 21581 . . 3 (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾)
108, 9eqeqan12d 2841 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ((topGen‘𝐽) = (topGen‘𝐾) ↔ 𝐽 = 𝐾))
112, 7, 103bitr3d 312 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ([𝐽] = [𝐾] 𝐽 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  cin 3918   class class class wbr 5052  ccnv 5541  cfv 6343   Er wer 8282  [cec 8283  topGenctg 16711  Topctop 21501  Fnecfne 33741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fv 6351  df-er 8285  df-ec 8287  df-topgen 16717  df-top 21502  df-fne 33742
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator