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Theorem brrelex12 5568
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 5526 . . . . 5 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 219 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
32ssbrd 5073 . . 3 (Rel 𝑅 → (𝐴𝑅𝐵𝐴(V × V)𝐵))
43imp 410 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴(V × V)𝐵)
5 brxp 5565 . 2 (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
64, 5sylib 221 1 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2111  Vcvv 3441  wss 3881   class class class wbr 5030   × cxp 5517  Rel wrel 5524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526
This theorem is referenced by:  brrelex1  5569  brrelex2  5570  brrelex12i  5571  relbrcnvg  5935  brovex  7871  ersym  8284  relelec  8317  fpwwe2lem2  10043  fpwwelem  10056  cofuval2  17149  isnat  17209  pslem  17808  frgpuplem  18890  perpln1  26504  perpln2  26505  poprelb  44036
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