Theorem brrelex12 5189

Description: A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (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 5150	. . . . 5 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 206	. . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3	2	ssbrd 4728	. . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐴(V × V)𝐵))
4	3	imp 444	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴(V × V)𝐵)
5		brxp 5181	. 2 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	4, 5	sylib 208	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607   class class class wbr 4685   × cxp 5141  Rel wrel 5148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150

This theorem is referenced by:  brrelex  5190  brrelex2  5191  nprrel12  5194  relbrcnvg  5539  ovprc  6723  oprabv  6745  brovex  7393  ersym  7799  relelec  7830  encv  8005  fsuppunbi  8337  fpwwe2lem2  9492  fpwwelem  9505  brfi1uzind  13318  isstruct2  15914  brssc  16521  cofuval2  16594  isfull  16617  isfth  16621  isnat  16654  pslem  17253  frgpuplem  18231  dvdsr  18692  ulmval  24179  perpln1  25650  perpln2  25651  opelco3  31802  rngoablo2  33838  aovprc  41589  aovrcl  41590  nelbrim  41616