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Theorem fneer 36336
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 6907 . 2 (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦))
2 fneval.1 . . . . . 6 = (Fne ∩ Fne)
3 inss1 4245 . . . . . 6 (Fne ∩ Fne) ⊆ Fne
42, 3eqsstri 4030 . . . . 5 ⊆ Fne
5 fnerel 36321 . . . . 5 Rel Fne
6 relss 5794 . . . . 5 ( ⊆ Fne → (Rel Fne → Rel ))
74, 5, 6mp2 9 . . . 4 Rel
8 dfrel4v 6212 . . . 4 (Rel = {⟨𝑥, 𝑦⟩ ∣ 𝑥 𝑦})
97, 8mpbi 230 . . 3 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 𝑦}
102fneval 36335 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)))
1110el2v 3485 . . . 4 (𝑥 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))
1211opabbii 5215 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑥 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
139, 12eqtri 2763 . 2 = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
141, 13eqer 8780 1 Er V
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  Vcvv 3478  cin 3962  wss 3963   class class class wbr 5148  {copab 5210  ccnv 5688  Rel wrel 5694  cfv 6563   Er wer 8741  topGenctg 17484  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-iota 6516  df-fun 6565  df-fv 6571  df-er 8744  df-topgen 17490  df-fne 36320
This theorem is referenced by:  topfneec  36338  topfneec2  36339
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