Theorem brrelex12i 5669

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 5666	. 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 2111  Vcvv 3436   class class class wbr 5089  Rel wrel 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621

This theorem is referenced by:  nprrel12  5672  vtoclr  5677  relbrcnvg  6053  ovprc  7384  oprabv  7406  encv  8877  brdomi  8882  domssl  8920  fsuppimp  9252  fsuppunbi  9273  brttrcl  9603  brfi1uzind  14415  brfi1indALT  14417  isstruct2  17060  brssc  17721  isfull  17819  isfth  17823  dvdsr  20280  ulmval  26316  subgrv  29248  vcex  30558  opelco3  35819  bj-ideqgALT  37202  bj-idreseqb  37207  bj-ideqg1ALT  37209  rngoablo2  37948  aovprc  47287  aovrcl  47288  nelbrim  47374  linindsv  48545  func1st  49177  func2nd  49178  oppfval  49236  upfval3  49278  prcofval  49478