Theorem brrelex12i 5693

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 5690	. 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 3447   class class class wbr 5107  Rel wrel 5643

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 5251  ax-nul 5261  ax-pr 5387

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645

This theorem is referenced by:  nprrel12  5696  vtoclr  5701  relbrcnvg  6076  ovprc  7425  oprabv  7449  encv  8926  brdomi  8931  domssl  8969  fsuppimp  9319  fsuppunbi  9340  brttrcl  9666  brfi1uzind  14473  brfi1indALT  14475  isstruct2  17119  brssc  17776  isfull  17874  isfth  17878  dvdsr  20271  ulmval  26289  subgrv  29197  vcex  30507  opelco3  35762  bj-ideqgALT  37146  bj-idreseqb  37151  bj-ideqg1ALT  37153  rngoablo2  37903  aovprc  47186  aovrcl  47187  nelbrim  47273  linindsv  48431  func1st  49063  func2nd  49064  oppfval  49122  upfval3  49164  prcofval  49364