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Theorem erexb 6131
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 4596 . . 3 (𝑅 ∈ V → dom 𝑅 ∈ V)
2 erdm 6116 . . . 4 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
32eleq1d 2106 . . 3 (𝑅 Er 𝐴 → (dom 𝑅 ∈ V ↔ 𝐴 ∈ V))
41, 3syl5ib 143 . 2 (𝑅 Er 𝐴 → (𝑅 ∈ V → 𝐴 ∈ V))
5 erex 6130 . 2 (𝑅 Er 𝐴 → (𝐴 ∈ V → 𝑅 ∈ V))
64, 5impbid 120 1 (𝑅 Er 𝐴 → (𝑅 ∈ V ↔ 𝐴 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wcel 1393  Vcvv 2557  dom cdm 4345   Er wer 6103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-er 6106
This theorem is referenced by: (None)
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