Theorem brrelex12i 5732

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 5729	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1, 2	mpan 689	1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107  Vcvv 3475   class class class wbr 5149  Rel wrel 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684

This theorem is referenced by:  nprrel12  5735  vtoclr  5740  relbrcnvg  6105  ovprc  7447  oprabv  7469  encv  8947  brdomi  8954  domssl  8994  fsuppimp  9368  fsuppunbi  9384  brttrcl  9708  brfi1uzind  14459  brfi1indALT  14461  isstruct2  17082  brssc  17761  isfull  17861  isfth  17865  dvdsr  20176  ulmval  25892  subgrv  28527  vcex  29831  opelco3  34746  bj-ideqgALT  36039  bj-idreseqb  36044  bj-ideqg1ALT  36046  rngoablo2  36777  aovprc  45896  aovrcl  45897  nelbrim  45983  isisomgr  46492  linindsv  47126