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Theorem fneer 33809
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 6649 . 2 (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦))
2 fneval.1 . . . . . 6 = (Fne ∩ Fne)
3 inss1 4158 . . . . . 6 (Fne ∩ Fne) ⊆ Fne
42, 3eqsstri 3952 . . . . 5 ⊆ Fne
5 fnerel 33794 . . . . 5 Rel Fne
6 relss 5624 . . . . 5 ( ⊆ Fne → (Rel Fne → Rel ))
74, 5, 6mp2 9 . . . 4 Rel
8 dfrel4v 6018 . . . 4 (Rel = {⟨𝑥, 𝑦⟩ ∣ 𝑥 𝑦})
97, 8mpbi 233 . . 3 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 𝑦}
102fneval 33808 . . . . 5 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)))
1110el2v 3451 . . . 4 (𝑥 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))
1211opabbii 5100 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑥 𝑦} = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
139, 12eqtri 2824 . 2 = {⟨𝑥, 𝑦⟩ ∣ (topGen‘𝑥) = (topGen‘𝑦)}
141, 13eqer 8311 1 Er V
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  Vcvv 3444  cin 3883  wss 3884   class class class wbr 5033  {copab 5095  ccnv 5522  Rel wrel 5528  cfv 6328   Er wer 8273  topGenctg 16706  Fnecfne 33792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-er 8276  df-topgen 16712  df-fne 33793
This theorem is referenced by:  topfneec  33811  topfneec2  33812
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