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Theorem erexb 6420
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 4771 . . 3 (𝑅 ∈ V → dom 𝑅 ∈ V)
2 erdm 6405 . . . 4 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
32eleq1d 2184 . . 3 (𝑅 Er 𝐴 → (dom 𝑅 ∈ V ↔ 𝐴 ∈ V))
41, 3syl5ib 153 . 2 (𝑅 Er 𝐴 → (𝑅 ∈ V → 𝐴 ∈ V))
5 erex 6419 . 2 (𝑅 Er 𝐴 → (𝐴 ∈ V → 𝑅 ∈ V))
64, 5impbid 128 1 (𝑅 Er 𝐴 → (𝑅 ∈ V ↔ 𝐴 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1463  Vcvv 2658  dom cdm 4507   Er wer 6392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-dm 4517  df-rn 4518  df-er 6395
This theorem is referenced by: (None)
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