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Theorem topfneec2 36357
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 36353 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 𝐾 ↔ (topGen‘𝐽) = (topGen‘𝐾)))
31fneer 36354 . . . 4 Er V
43a1i 11 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → Er V)
5 elex 3501 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
65adantr 480 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → 𝐽 ∈ V)
74, 6erth 8796 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 𝐾 ↔ [𝐽] = [𝐾] ))
8 tgtop 22980 . . 3 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
9 tgtop 22980 . . 3 (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾)
108, 9eqeqan12d 2751 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ((topGen‘𝐽) = (topGen‘𝐾) ↔ 𝐽 = 𝐾))
112, 7, 103bitr3d 309 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ([𝐽] = [𝐾] 𝐽 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950   class class class wbr 5143  ccnv 5684  cfv 6561   Er wer 8742  [cec 8743  topGenctg 17482  Topctop 22899  Fnecfne 36337
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-er 8745  df-ec 8747  df-topgen 17488  df-top 22900  df-fne 36338
This theorem is referenced by: (None)
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