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Theorem fneer 32323
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 6178 . 2 (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦))
2 fneval.1 . . . . . 6 = (Fne ∩ Fne)
3 inss1 3825 . . . . . 6 (Fne ∩ Fne) ⊆ Fne
42, 3eqsstri 3627 . . . . 5 ⊆ Fne
5 fnerel 32308 . . . . 5 Rel Fne
6 relss 5196 . . . . 5 ( ⊆ Fne → (Rel Fne → Rel ))
74, 5, 6mp2 9 . . . 4 Rel
8 dfrel4v 5572 . . . 4 (Rel = {⟨𝑥, 𝑦⟩ ∣ 𝑥 𝑦})
97, 8mpbi 220 . . 3 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 𝑦}
10 vex 3198 . . . . 5 𝑥 ∈ V
11 vex 3198 . . . . 5 𝑦 ∈ V
122fneval 32322 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)))
1310, 11, 12mp2an 707 . . . 4 (𝑥 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))
1413opabbii 4708 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑥 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
159, 14eqtri 2642 . 2 = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
161, 15eqer 7762 1 Er V
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1481  wcel 1988  Vcvv 3195  cin 3566  wss 3567   class class class wbr 4644  {copab 4703  ccnv 5103  Rel wrel 5109  cfv 5876   Er wer 7724  topGenctg 16079  Fnecfne 32306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-er 7727  df-topgen 16085  df-fne 32307
This theorem is referenced by:  topfneec  32325  topfneec2  32326
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