Theorem brrelex12 5693

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 5648	. . . . 5 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 216	. . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3	2	ssbrd 5153	. . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐴(V × V)𝐵))
4	3	imp 406	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴(V × V)𝐵)
5		brxp 5690	. 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 3450   ⊆ wss 3917   class class class wbr 5110   × cxp 5639  Rel wrel 5646

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648

This theorem is referenced by:  brrelex1  5694  brrelex2  5695  brrelex12i  5696  relbrcnvg  6079  brovex  8204  ersym  8686  relelec  8721  fpwwe2lem2  10592  fpwwelem  10605  cofuval2  17856  isnat  17919  pslem  18538  frgpuplem  19709  perpln1  28644  perpln2  28645  poprelb  47529  precofval3  49364