Theorem brrelex12i 5689

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 5686	. 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 3442   class class class wbr 5100  Rel wrel 5639

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 5245  ax-pr 5381

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 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5640  df-rel 5641

This theorem is referenced by:  nprrel12  5692  vtoclr  5697  relbrcnvg  6074  ovprc  7408  oprabv  7430  encv  8905  brdomi  8910  domssl  8949  fsuppimp  9285  fsuppunbi  9306  brttrcl  9636  brfi1uzind  14445  brfi1indALT  14447  isstruct2  17090  brssc  17752  isfull  17850  isfth  17854  dvdsr  20315  ulmval  26362  subgrv  29361  vcex  30672  opelco3  35997  bj-ideqgALT  37440  bj-idreseqb  37445  bj-ideqg1ALT  37447  rngoablo2  38189  aovprc  47577  aovrcl  47578  nelbrim  47664  linindsv  48834  func1st  49465  func2nd  49466  oppfval  49524  upfval3  49566  prcofval  49766