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Theorem erexb 6317
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 4697 . . 3 (𝑅 ∈ V → dom 𝑅 ∈ V)
2 erdm 6302 . . . 4 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
32eleq1d 2156 . . 3 (𝑅 Er 𝐴 → (dom 𝑅 ∈ V ↔ 𝐴 ∈ V))
41, 3syl5ib 152 . 2 (𝑅 Er 𝐴 → (𝑅 ∈ V → 𝐴 ∈ V))
5 erex 6316 . 2 (𝑅 Er 𝐴 → (𝐴 ∈ V → 𝑅 ∈ V))
64, 5impbid 127 1 (𝑅 Er 𝐴 → (𝑅 ∈ V ↔ 𝐴 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wcel 1438  Vcvv 2619  dom cdm 4438   Er wer 6289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-er 6292
This theorem is referenced by: (None)
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