Theorem erex 8695

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 8694	. . 3 ⊢ (𝑅 Er 𝐴 → 𝑅 ⊆ (𝐴 × 𝐴))
2		sqxpexg 7731	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
3		ssexg 5278	. . 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 412	1 ⊢ (𝑅 Er 𝐴 → (𝐴 ∈ 𝑉 → 𝑅 ∈ V))

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3447   ⊆ wss 3914   × cxp 5636   Er wer 8668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-er 8671

This theorem is referenced by:  erexb  8696  qliftlem  8771  qshash  15793  qusaddvallem  17514  qusaddflem  17515  qusaddval  17516  qusaddf  17517  qusmulval  17518  qusmulf  17519  qusgrp2  18990  efgrelexlemb  19680  efgcpbllemb  19685  frgpuplem  19702  qusrng  20089  qusring2  20243  vitalilem2  25510  vitalilem3  25511  tgjustr  28401