Theorem brrelex12i 5680

Description: Two classes that are related by a binary relation are sets. Inference form. (Contributed by BJ, 3-Oct-2022.)

Hypothesis

Ref	Expression
brrelexi.1	⊢ Rel 𝑅

Assertion

Ref	Expression
brrelex12i	⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem brrelex12i

Step	Hyp	Ref	Expression
1		brrelexi.1	. 2 ⊢ Rel 𝑅
2		brrelex12 5677	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1, 2	mpan 691	1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Vcvv 3441   class class class wbr 5099  Rel wrel 5630

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632

This theorem is referenced by:  nprrel12  5683  vtoclr  5688  relbrcnvg  6065  ovprc  7398  oprabv  7420  encv  8895  brdomi  8900  domssl  8939  fsuppimp  9275  fsuppunbi  9296  brttrcl  9626  brfi1uzind  14435  brfi1indALT  14437  isstruct2  17080  brssc  17742  isfull  17840  isfth  17844  dvdsr  20302  ulmval  26349  subgrv  29347  vcex  30657  opelco3  35971  bj-ideqgALT  37365  bj-idreseqb  37370  bj-ideqg1ALT  37372  rngoablo2  38112  aovprc  47501  aovrcl  47502  nelbrim  47588  linindsv  48758  func1st  49389  func2nd  49390  oppfval  49448  upfval3  49490  prcofval  49690