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Theorem brrelex12 5079
 Description: A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
brrelex12 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem brrelex12
StepHypRef Expression
1 df-rel 5045 . . . . 5 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 205 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
32ssbrd 4626 . . 3 (Rel 𝑅 → (𝐴𝑅𝐵𝐴(V × V)𝐵))
43imp 444 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴(V × V)𝐵)
5 brxp 5071 . 2 (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
64, 5sylib 207 1 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   × cxp 5036  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045 This theorem is referenced by:  brrelex  5080  brrelex2  5081  relbrcnvg  5423  ovprc  6581  oprabv  6601  brovex  7235  ersym  7641  relelec  7674  encv  7849  fsuppunbi  8179  fpwwe2lem2  9333  fpwwelem  9346  brfi1uzind  13135  brfi1uzindOLD  13141  isstruct2  15704  brssc  16297  cofuval2  16370  isfull  16393  isfth  16397  isnat  16430  pslem  17029  frgpuplem  18008  dvdsr  18469  ulmval  23938  perpln1  25405  perpln2  25406  iseupa  26492  opelco3  30923  rngoablo2  32878  aovprc  39917  aovrcl  39918
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