Theorem brrelex12 5671

Description: Two classes related by a binary relation are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
brrelex12	⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem brrelex12

Step	Hyp	Ref	Expression
1		df-rel 5626	. . . . 5 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 216	. . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3	2	ssbrd 5135	. . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐴(V × V)𝐵))
4	3	imp 406	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴(V × V)𝐵)
5		brxp 5668	. 2 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	4, 5	sylib 218	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3436   ⊆ wss 3903   class class class wbr 5092   × cxp 5617  Rel wrel 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626

This theorem is referenced by:  brrelex1  5672  brrelex2  5673  brrelex12i  5674  relbrcnvg  6056  brovex  8155  ersym  8637  relelec  8672  fpwwe2lem2  10526  fpwwelem  10539  cofuval2  17794  isnat  17857  pslem  18478  frgpuplem  19651  perpln1  28659  perpln2  28660  poprelb  47528  precofval3  49376