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Theorem brrelex12 5728
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 5683 . . . . 5 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 215 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
32ssbrd 5191 . . 3 (Rel 𝑅 → (𝐴𝑅𝐵𝐴(V × V)𝐵))
43imp 407 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴(V × V)𝐵)
5 brxp 5725 . 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 396  wcel 2106  Vcvv 3474  wss 3948   class class class wbr 5148   × cxp 5674  Rel wrel 5681
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683
This theorem is referenced by:  brrelex1  5729  brrelex2  5730  brrelex12i  5731  relbrcnvg  6104  brovex  8209  ersym  8717  relelec  8750  fpwwe2lem2  10629  fpwwelem  10642  cofuval2  17839  isnat  17900  pslem  18527  frgpuplem  19642  perpln1  27999  perpln2  28000  poprelb  46271
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