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

Theorem brel 5037
Description: Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
brel.1  |-  R  C_  ( C  X.  D
)
Assertion
Ref Expression
brel  |-  ( A R B  ->  ( A  e.  C  /\  B  e.  D )
)

Proof of Theorem brel
StepHypRef Expression
1 brel.1 . . 3  |-  R  C_  ( C  X.  D
)
21ssbri 4481 . 2  |-  ( A R B  ->  A
( C  X.  D
) B )
3 brxp 5019 . 2  |-  ( A ( C  X.  D
) B  <->  ( A  e.  C  /\  B  e.  D ) )
42, 3sylib 196 1  |-  ( A R B  ->  ( A  e.  C  /\  B  e.  D )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823    C_ wss 3461   class class class wbr 4439    X. cxp 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994
This theorem is referenced by:  brab2a  5038  brab2ga  5064  soirri  5381  sotri  5382  sotri2  5384  sotri3  5385  ndmovord  6438  ndmovordi  6439  swoer  7331  brecop2  7397  ecopovsym  7405  ecopovtrn  7406  hartogslem1  7959  nlt1pi  9273  indpi  9274  nqerf  9297  ordpipq  9309  lterpq  9337  ltexnq  9342  ltbtwnnq  9345  ltrnq  9346  prnmadd  9364  genpcd  9373  nqpr  9381  1idpr  9396  ltexprlem4  9406  ltexpri  9410  ltaprlem  9411  prlem936  9414  reclem2pr  9415  reclem3pr  9416  reclem4pr  9417  suplem1pr  9419  suplem2pr  9420  supexpr  9421  recexsrlem  9469  addgt0sr  9470  mulgt0sr  9471  mappsrpr  9474  map2psrpr  9476  supsrlem  9477  supsr  9478  ltresr  9506  dfle2  11356  dflt2  11357  dvdszrcl  14075  letsr  16056  hmphtop  20445  vcex  25671  brtxp2  29759  brpprod3a  29764
  Copyright terms: Public domain W3C validator