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Theorem erexb 8725
Description: An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
erexb (𝑅 Er 𝐴 → (𝑅 ∈ V ↔ 𝐴 ∈ V))

Proof of Theorem erexb
StepHypRef Expression
1 dmexg 7888 . . 3 (𝑅 ∈ V → dom 𝑅 ∈ V)
2 erdm 8710 . . . 4 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
32eleq1d 2810 . . 3 (𝑅 Er 𝐴 → (dom 𝑅 ∈ V ↔ 𝐴 ∈ V))
41, 3imbitrid 243 . 2 (𝑅 Er 𝐴 → (𝑅 ∈ V → 𝐴 ∈ V))
5 erex 8724 . 2 (𝑅 Er 𝐴 → (𝐴 ∈ V → 𝑅 ∈ V))
64, 5impbid 211 1 (𝑅 Er 𝐴 → (𝑅 ∈ V ↔ 𝐴 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  Vcvv 3466  dom cdm 5667   Er wer 8697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-er 8700
This theorem is referenced by:  prtex  38253
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