Theorem brrelex12i 5733

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 5730	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1, 2	mpan 688	1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2098  Vcvv 3461   class class class wbr 5149  Rel wrel 5683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685

This theorem is referenced by:  nprrel12  5736  vtoclr  5741  relbrcnvg  6110  ovprc  7457  oprabv  7480  encv  8972  brdomi  8979  domssl  9019  fsuppimp  9394  fsuppunbi  9414  brttrcl  9738  brfi1uzind  14495  brfi1indALT  14497  isstruct2  17121  brssc  17800  isfull  17902  isfth  17906  dvdsr  20313  ulmval  26361  subgrv  29155  vcex  30460  opelco3  35501  bj-ideqgALT  36768  bj-idreseqb  36773  bj-ideqg1ALT  36775  rngoablo2  37513  aovprc  46706  aovrcl  46707  nelbrim  46793  linindsv  47699