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Theorem brrelex12 5734
Description: Two classes related by a binary relation are sets. (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 5689 . . . . 5 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 215 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
32ssbrd 5196 . . 3 (Rel 𝑅 → (𝐴𝑅𝐵𝐴(V × V)𝐵))
43imp 405 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴(V × V)𝐵)
5 brxp 5731 . 2 (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
64, 5sylib 217 1 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2099  Vcvv 3462  wss 3947   class class class wbr 5153   × cxp 5680  Rel wrel 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689
This theorem is referenced by:  brrelex1  5735  brrelex2  5736  brrelex12i  5737  relbrcnvg  6115  brovex  8237  ersym  8746  relelec  8781  fpwwe2lem2  10675  fpwwelem  10688  cofuval2  17906  isnat  17970  pslem  18597  frgpuplem  19770  perpln1  28637  perpln2  28638  poprelb  47096
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