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Theorem erex 6296
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 6295 . . 3 (𝑅 Er 𝐴𝑅 ⊆ (𝐴 × 𝐴))
2 xpexg 4540 . . . 4 ((𝐴𝑉𝐴𝑉) → (𝐴 × 𝐴) ∈ V)
32anidms 389 . . 3 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
4 ssexg 3970 . . 3 ((𝑅 ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → 𝑅 ∈ V)
51, 3, 4syl2an 283 . 2 ((𝑅 Er 𝐴𝐴𝑉) → 𝑅 ∈ V)
65ex 113 1 (𝑅 Er 𝐴 → (𝐴𝑉𝑅 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1438  Vcvv 2619  wss 2997   × cxp 4426   Er wer 6269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-dm 4438  df-rn 4439  df-er 6272
This theorem is referenced by:  erexb  6297  qliftlem  6350
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