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Theorem fneer 36354
Description: Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
fneval.1 = (Fne ∩ Fne)
Assertion
Ref Expression
fneer Er V

Proof of Theorem fneer
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6906 . 2 (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦))
2 fneval.1 . . . . . 6 = (Fne ∩ Fne)
3 inss1 4237 . . . . . 6 (Fne ∩ Fne) ⊆ Fne
42, 3eqsstri 4030 . . . . 5 ⊆ Fne
5 fnerel 36339 . . . . 5 Rel Fne
6 relss 5791 . . . . 5 ( ⊆ Fne → (Rel Fne → Rel ))
74, 5, 6mp2 9 . . . 4 Rel
8 dfrel4v 6210 . . . 4 (Rel = {⟨𝑥, 𝑦⟩ ∣ 𝑥 𝑦})
97, 8mpbi 230 . . 3 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 𝑦}
102fneval 36353 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)))
1110el2v 3487 . . . 4 (𝑥 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))
1211opabbii 5210 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑥 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
139, 12eqtri 2765 . 2 = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
141, 13eqer 8781 1 Er V
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  Vcvv 3480  cin 3950  wss 3951   class class class wbr 5143  {copab 5205  ccnv 5684  Rel wrel 5690  cfv 6561   Er wer 8742  topGenctg 17482  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-iota 6514  df-fun 6563  df-fv 6569  df-er 8745  df-topgen 17488  df-fne 36338
This theorem is referenced by:  topfneec  36356  topfneec2  36357
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