Theorem brrelex12i 5730

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 5727	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1, 2	mpan 686	1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2104  Vcvv 3472   class class class wbr 5147  Rel wrel 5680

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682

This theorem is referenced by:  nprrel12  5733  vtoclr  5738  relbrcnvg  6103  ovprc  7449  oprabv  7471  encv  8949  brdomi  8956  domssl  8996  fsuppimp  9370  fsuppunbi  9386  brttrcl  9710  brfi1uzind  14463  brfi1indALT  14465  isstruct2  17086  brssc  17765  isfull  17865  isfth  17869  dvdsr  20253  ulmval  26128  subgrv  28794  vcex  30098  opelco3  35050  bj-ideqgALT  36342  bj-idreseqb  36347  bj-ideqg1ALT  36349  rngoablo2  37080  aovprc  46194  aovrcl  46195  nelbrim  46281  isisomgr  46790  linindsv  47213