Theorem brrelex12i 5739

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 5736	. 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 2107  Vcvv 3479   class class class wbr 5142  Rel wrel 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691

This theorem is referenced by:  nprrel12  5742  vtoclr  5747  relbrcnvg  6122  ovprc  7470  oprabv  7494  encv  8994  brdomi  9000  domssl  9039  fsuppimp  9409  fsuppunbi  9430  brttrcl  9754  brfi1uzind  14548  brfi1indALT  14550  isstruct2  17187  brssc  17859  isfull  17958  isfth  17962  dvdsr  20363  ulmval  26424  subgrv  29288  vcex  30598  opelco3  35776  bj-ideqgALT  37160  bj-idreseqb  37165  bj-ideqg1ALT  37167  rngoablo2  37917  aovprc  47205  aovrcl  47206  nelbrim  47292  linindsv  48367  upfval3  48953