Theorem brrelex12i 5683

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 5680	. 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 3430   class class class wbr 5086  Rel wrel 5633

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 5232  ax-pr 5374

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5634  df-rel 5635

This theorem is referenced by:  nprrel12  5686  vtoclr  5691  relbrcnvg  6068  ovprc  7402  oprabv  7424  encv  8898  brdomi  8903  domssl  8942  fsuppimp  9278  fsuppunbi  9299  brttrcl  9631  brfi1uzind  14467  brfi1indALT  14469  isstruct2  17116  brssc  17778  isfull  17876  isfth  17880  dvdsr  20339  ulmval  26364  subgrv  29359  vcex  30670  opelco3  35979  bj-ideqgALT  37494  bj-idreseqb  37499  bj-ideqg1ALT  37501  rngoablo2  38252  aovprc  47656  aovrcl  47657  nelbrim  47743  linindsv  48941  func1st  49572  func2nd  49573  oppfval  49631  upfval3  49673  prcofval  49873