Theorem brrelex2 5691

Description: If two classes are related by a binary relation, then the second class is a set. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
brrelex2	⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)

Proof of Theorem brrelex2

Step	Hyp	Ref	Expression
1		brrelex12 5689	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2	1	simprd 496	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  Vcvv 3446   class class class wbr 5110  Rel wrel 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645

This theorem is referenced by:  brrelex2i  5694  releldm  5904  relelrn  5905  elrelimasn  6042  funbrfv  6898  relbrtpos  8173  ertr  8670  erth  8704  pslem  18475  opeldifid  31584  eqvreltr  37142  eqvrelth  37146  frege124d  42155  frege133d  42159  climfv  44052  funbrafv2  45599