Theorem brrelex2 5676

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 5674	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2	1	simprd 495	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  Vcvv 3438   class class class wbr 5096  Rel wrel 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629

This theorem is referenced by:  brrelex2i  5679  releldm  5891  relelrn  5892  elrelimasn  6043  funbrfv  6880  relbrtpos  8177  ertr  8648  erth  8687  fsuppss  9284  pslem  18493  opeldifid  32623  eqvreltr  38803  eqvrelth  38807  frege124d  43944  frege133d  43948  climfv  45877  funbrafv2  47435