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Theorem topfneec2 36339
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 36335 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 𝐾 ↔ (topGen‘𝐽) = (topGen‘𝐾)))
31fneer 36336 . . . 4 Er V
43a1i 11 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → Er V)
5 elex 3499 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
65adantr 480 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → 𝐽 ∈ V)
74, 6erth 8795 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 𝐾 ↔ [𝐽] = [𝐾] ))
8 tgtop 22996 . . 3 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
9 tgtop 22996 . . 3 (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾)
108, 9eqeqan12d 2749 . 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 1537  wcel 2106  Vcvv 3478  cin 3962   class class class wbr 5148  ccnv 5688  cfv 6563   Er wer 8741  [cec 8742  topGenctg 17484  Topctop 22915  Fnecfne 36319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-er 8744  df-ec 8746  df-topgen 17490  df-top 22916  df-fne 36320
This theorem is referenced by: (None)
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