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Theorem erex 7935
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 7934 . . 3 (𝑅 Er 𝐴𝑅 ⊆ (𝐴 × 𝐴))
2 sqxpexg 7128 . . 3 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
3 ssexg 4956 . . 3 ((𝑅 ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → 𝑅 ∈ V)
41, 2, 3syl2an 495 . 2 ((𝑅 Er 𝐴𝐴𝑉) → 𝑅 ∈ V)
54ex 449 1 (𝑅 Er 𝐴 → (𝐴𝑉𝑅 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139  Vcvv 3340  wss 3715   × cxp 5264   Er wer 7908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-er 7911
This theorem is referenced by:  erexb  7936  qliftlem  7995  qshash  14758  qusaddvallem  16413  qusaddflem  16414  qusaddval  16415  qusaddf  16416  qusmulval  16417  qusmulf  16418  qusgrp2  17734  efgrelexlemb  18363  efgcpbllemb  18368  frgpuplem  18385  qusring2  18820  vitalilem2  23577  vitalilem3  23578
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