MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brrelex12 Structured version   Unicode version

Theorem brrelex12 4978
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  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )

Proof of Theorem brrelex12
StepHypRef Expression
1 df-rel 4947 . . . . 5  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 194 . . . 4  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32ssbrd 4433 . . 3  |-  ( Rel 
R  ->  ( A R B  ->  A ( _V  X.  _V ) B ) )
43imp 427 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A
( _V  X.  _V ) B )
5 brxp 4971 . 2  |-  ( A ( _V  X.  _V ) B  <->  ( A  e. 
_V  /\  B  e.  _V ) )
64, 5sylib 196 1  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1840   _Vcvv 3056    C_ wss 3411   class class class wbr 4392    X. cxp 4938   Rel wrel 4945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947
This theorem is referenced by:  brrelex  4979  brrelex2  4980  relbrcnvg  5315  ovprc  6262  oprabv  6280  brovex  6905  ersym  7278  relelec  7307  encv  7480  fsuppunbi  7802  fpwwe2lem2  8958  fpwwelem  8971  isstruct2  14740  brssc  15317  cofuval2  15390  isfull  15413  isfth  15417  isnat  15450  pslem  16050  frgpuplem  17004  dvdsr  17505  tpsexOLD  19602  ulmval  22957  perpln1  24367  perpln2  24368  iseupa  25264  rngoablo2  25719  opelco3  29874  aovprc  37608  aovrcl  37609
  Copyright terms: Public domain W3C validator