Theorem brrelex12i 5674

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 5671	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1, 2	mpan 690	1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3436   class class class wbr 5092  Rel wrel 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626

This theorem is referenced by:  nprrel12  5677  vtoclr  5682  relbrcnvg  6056  ovprc  7387  oprabv  7409  encv  8880  brdomi  8885  domssl  8923  fsuppimp  9258  fsuppunbi  9279  brttrcl  9609  brfi1uzind  14415  brfi1indALT  14417  isstruct2  17060  brssc  17721  isfull  17819  isfth  17823  dvdsr  20247  ulmval  26287  subgrv  29219  vcex  30526  opelco3  35768  bj-ideqgALT  37152  bj-idreseqb  37157  bj-ideqg1ALT  37159  rngoablo2  37909  aovprc  47192  aovrcl  47193  nelbrim  47279  linindsv  48450  func1st  49082  func2nd  49083  oppfval  49141  upfval3  49183  prcofval  49383