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Theorem brrelex2i 4590
 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 4587 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
31, 2mpan 421 1 (𝐴𝑅𝐵𝐵 ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1481  Vcvv 2689   class class class wbr 3936  Rel wrel 4551 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-rel 4553 This theorem is referenced by:  vtoclr  4594  brdomi  6650  xpdom2  6732  xpdom1g  6734  mapdom1g  6748  djudom  6985  difinfsn  6992  enomnilem  7017  enmkvlem  7042  enwomnilem  7049  djuenun  7084  aprcl  8431  hashinfom  10555  clim  11081  ntrivcvgap0  11349
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