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Theorem erex 7711
Description: An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
erex (𝑅 Er 𝐴 → (𝐴𝑉𝑅 ∈ V))

Proof of Theorem erex
StepHypRef Expression
1 erssxp 7710 . . 3 (𝑅 Er 𝐴𝑅 ⊆ (𝐴 × 𝐴))
2 sqxpexg 6916 . . 3 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
3 ssexg 4764 . . 3 ((𝑅 ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → 𝑅 ∈ V)
41, 2, 3syl2an 494 . 2 ((𝑅 Er 𝐴𝐴𝑉) → 𝑅 ∈ V)
54ex 450 1 (𝑅 Er 𝐴 → (𝐴𝑉𝑅 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  Vcvv 3186  wss 3555   × cxp 5072   Er wer 7684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085  df-er 7687
This theorem is referenced by:  erexb  7712  qliftlem  7773  qshash  14484  qusaddvallem  16132  qusaddflem  16133  qusaddval  16134  qusaddf  16135  qusmulval  16136  qusmulf  16137  qusgrp2  17454  efgrelexlemb  18084  efgcpbllemb  18089  frgpuplem  18106  qusring2  18541  vitalilem2  23284  vitalilem3  23285
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