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Theorem brrelex2i 4688
Description: The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
brrelexi.1 Rel 𝑅
Assertion
Ref Expression
brrelex2i (𝐴𝑅𝐵𝐵 ∈ V)

Proof of Theorem brrelex2i
StepHypRef Expression
1 brrelexi.1 . 2 Rel 𝑅
2 brrelex2 4685 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
31, 2mpan 424 1 (𝐴𝑅𝐵𝐵 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2160  Vcvv 2752   class class class wbr 4018  Rel wrel 4649
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-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651
This theorem is referenced by:  vtoclr  4692  brdomi  6775  xpdom2  6857  xpdom1g  6859  mapdom1g  6875  djudom  7122  difinfsn  7129  enomnilem  7166  enmkvlem  7189  enwomnilem  7197  djuenun  7241  aprcl  8633  hashinfom  10790  clim  11321  ntrivcvgap0  11589
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