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Theorem brrelex2i 4666
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 4663 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)
31, 2mpan 424 1 (𝐴𝑅𝐵𝐵 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  Vcvv 2737   class class class wbr 4000  Rel wrel 4627
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629
This theorem is referenced by:  vtoclr  4670  brdomi  6742  xpdom2  6824  xpdom1g  6826  mapdom1g  6840  djudom  7085  difinfsn  7092  enomnilem  7129  enmkvlem  7152  enwomnilem  7160  djuenun  7204  aprcl  8580  hashinfom  10729  clim  11260  ntrivcvgap0  11528
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