Theorem erex 8715

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 8714	. . 3 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴))
2		sqxpexg 7750	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
3		ssexg 5291	. . 3 ⊢ ((𝑅 ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → 𝑅 ∈ V)
4	1, 2, 3	syl2an 607	. 2 ⊢ ((𝑅 Er 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑅 ∈ V)
5	4	ex 417	1 ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ 𝑉 → 𝑅 ∈ V))

Syntax hints:   → wi 4   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913   × cxp 5657   Er wer 8687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-er 8690

This theorem is referenced by:  erexb  8716  qliftlem  8792  qshash  15875  qusaddvallem  17601  qusaddflem  17602  qusaddval  17603  qusaddf  17604  qusmulval  17605  qusmulf  17606  qusgrp2  19120  efgrelexlemb  19816  efgcpbllemb  19821  frgpuplem  19838  qusrng  20254  qusring2  20412  vitalilem2  25733  vitalilem3  25734  tgjustr  28705