MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brrelex12 Structured version   Visualization version   GIF version

Theorem brrelex12 5729
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 5684 . . . . 5 (Rel 𝑅𝑅 ⊆ (V × V))
21biimpi 215 . . . 4 (Rel 𝑅𝑅 ⊆ (V × V))
32ssbrd 5192 . . 3 (Rel 𝑅 → (𝐴𝑅𝐵𝐴(V × V)𝐵))
43imp 406 . 2 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴(V × V)𝐵)
5 brxp 5726 . 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 395  wcel 2105  Vcvv 3473  wss 3949   class class class wbr 5149   × cxp 5675  Rel wrel 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684
This theorem is referenced by:  brrelex1  5730  brrelex2  5731  brrelex12i  5732  relbrcnvg  6105  brovex  8210  ersym  8718  relelec  8751  fpwwe2lem2  10630  fpwwelem  10643  cofuval2  17842  isnat  17903  pslem  18530  frgpuplem  19682  perpln1  28225  perpln2  28226  poprelb  46492
  Copyright terms: Public domain W3C validator