Theorem brrelex12i 5710

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 5707	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1, 2	mpan 703	1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  Vcvv 3454   class class class wbr 5106  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662

This theorem is used by:  nprrel12  5713  vtoclr  5718  relbrcnvg  6112  ovprc  7478  oprabv  7500  encv  9006  brdomi  9011  domssl  9050  fsuppimp  9385  fsuppunbi  9405  brttrcl  9738  brfi1uzind  14677  brfi1indALT  14679  isstruct2  17353  brssc  18015  isfull  18113  isfth  18117  dvdsr  20619  ulmval  26734  subgrv  29878  vcex  31207  opelco3  36716  bj-ideqgALT  38270  bj-idreseqb  38275  bj-ideqg1ALT  38277  rngoablo2  39034  aovprc  48461  aovrcl  48462  nelbrim  48548  linindsv  49760  func1st  50394  func2nd  50395  oppfval  50455  upfval3  50497  prcofval  50716