Theorem brrelex12i 5703

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 5700	. 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 2145  Vcvv 3450   class class class wbr 5103  Rel wrel 5653

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5249  ax-pr 5391

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5654  df-rel 5655

This theorem is used by:  nprrel12  5706  vtoclr  5711  relbrcnvg  6096  ovprc  7447  oprabv  7469  encv  8960  brdomi  8965  domssl  9004  fsuppimp  9338  fsuppunbi  9359  brttrcl  9692  brfi1uzind  14606  brfi1indALT  14608  isstruct2  17274  brssc  17936  isfull  18034  isfth  18038  dvdsr  20539  ulmval  26656  subgrv  29770  vcex  31099  opelco3  36455  bj-ideqgALT  37993  bj-idreseqb  37998  bj-ideqg1ALT  38000  rngoablo2  38757  aovprc  48174  aovrcl  48175  nelbrim  48261  linindsv  49473  func1st  50101  func2nd  50102  oppfval  50160  upfval3  50202  prcofval  50402