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Theorem erex 6613
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 6612 . . 3 (𝑅 Er 𝐴𝑅 ⊆ (𝐴 × 𝐴))
2 xpexg 4774 . . . 4 ((𝐴𝑉𝐴𝑉) → (𝐴 × 𝐴) ∈ V)
32anidms 397 . . 3 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
4 ssexg 4169 . . 3 ((𝑅 ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → 𝑅 ∈ V)
51, 3, 4syl2an 289 . 2 ((𝑅 Er 𝐴𝐴𝑉) → 𝑅 ∈ V)
65ex 115 1 (𝑅 Er 𝐴 → (𝐴𝑉𝑅 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2164  Vcvv 2760  wss 3154   × cxp 4658   Er wer 6586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-dm 4670  df-rn 4671  df-er 6589
This theorem is referenced by:  erexb  6614  qliftlem  6669  qusaddvallemg  12919  qusaddflemg  12920  qusaddval  12921  qusaddf  12922  qusmulval  12923  qusmulf  12924  qusgrp2  13186  eqgen  13300  qusrng  13457  qusring2  13565
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