Theorem brrelex12i 5686

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 5683	. 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 3429   class class class wbr 5085  Rel wrel 5636

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 2708  ax-sep 5231  ax-pr 5375

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638

This theorem is referenced by:  nprrel12  5689  vtoclr  5694  relbrcnvg  6070  ovprc  7405  oprabv  7427  encv  8901  brdomi  8906  domssl  8945  fsuppimp  9281  fsuppunbi  9302  brttrcl  9634  brfi1uzind  14470  brfi1indALT  14472  isstruct2  17119  brssc  17781  isfull  17879  isfth  17883  dvdsr  20342  ulmval  26345  subgrv  29339  vcex  30649  opelco3  35957  bj-ideqgALT  37472  bj-idreseqb  37477  bj-ideqg1ALT  37479  rngoablo2  38230  aovprc  47636  aovrcl  47637  nelbrim  47723  linindsv  48921  func1st  49552  func2nd  49553  oppfval  49611  upfval3  49653  prcofval  49853