Theorem erex 8729

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 → (𝐴 ∈ 𝑉 → 𝑅 ∈ V))

Proof of Theorem erex

Step	Hyp	Ref	Expression
1		erssxp 8728	. . 3 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴))
2		sqxpexg 7744	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
3		ssexg 5323	. . 3 ⊢ ((𝑅 ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → 𝑅 ∈ V)
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝑅 Er 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑅 ∈ V)
5	4	ex 413	1 ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ 𝑉 → 𝑅 ∈ V))

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3474   ⊆ wss 3948   × cxp 5674   Er wer 8702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-er 8705

This theorem is referenced by:  erexb  8730  qliftlem  8794  qshash  15775  qusaddvallem  17499  qusaddflem  17500  qusaddval  17501  qusaddf  17502  qusmulval  17503  qusmulf  17504  qusgrp2  18943  efgrelexlemb  19620  efgcpbllemb  19625  frgpuplem  19642  qusring2  20151  vitalilem2  25133  vitalilem3  25134  tgjustr  27763  qusrng  46766