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Theorem topfneec2 32947
 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 32943 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 𝐾 ↔ (topGen‘𝐽) = (topGen‘𝐾)))
31fneer 32944 . . . 4 Er V
43a1i 11 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → Er V)
5 elex 3414 . . . 4 (𝐽 ∈ Top → 𝐽 ∈ V)
65adantr 474 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → 𝐽 ∈ V)
74, 6erth 8075 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 𝐾 ↔ [𝐽] = [𝐾] ))
8 tgtop 21196 . . 3 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
9 tgtop 21196 . . 3 (𝐾 ∈ Top → (topGen‘𝐾) = 𝐾)
108, 9eqeqan12d 2794 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ((topGen‘𝐽) = (topGen‘𝐾) ↔ 𝐽 = 𝐾))
112, 7, 103bitr3d 301 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ([𝐽] = [𝐾] 𝐽 = 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ∩ cin 3791   class class class wbr 4888  ◡ccnv 5356  ‘cfv 6137   Er wer 8025  [cec 8026  topGenctg 16495  Topctop 21116  Fnecfne 32927 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fv 6145  df-er 8028  df-ec 8030  df-topgen 16501  df-top 21117  df-fne 32928 This theorem is referenced by: (None)
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