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Theorem erex 6535
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 6534 . . 3 (𝑅 Er 𝐴𝑅 ⊆ (𝐴 × 𝐴))
2 xpexg 4723 . . . 4 ((𝐴𝑉𝐴𝑉) → (𝐴 × 𝐴) ∈ V)
32anidms 395 . . 3 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
4 ssexg 4126 . . 3 ((𝑅 ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → 𝑅 ∈ V)
51, 3, 4syl2an 287 . 2 ((𝑅 Er 𝐴𝐴𝑉) → 𝑅 ∈ V)
65ex 114 1 (𝑅 Er 𝐴 → (𝐴𝑉𝑅 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2141  Vcvv 2730  wss 3121   × cxp 4607   Er wer 6508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-cnv 4617  df-dm 4619  df-rn 4620  df-er 6511
This theorem is referenced by:  erexb  6536  qliftlem  6589
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