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Theorem brel 4926
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 4254 . 2  |-  ( A R B  ->  A
( C  X.  D
) B )
3 brxp 4909 . 2  |-  ( A ( C  X.  D
) B  <->  ( A  e.  C  /\  B  e.  D ) )
42, 3sylib 189 1  |-  ( A R B  ->  ( A  e.  C  /\  B  e.  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    C_ wss 3320   class class class wbr 4212    X. cxp 4876
This theorem is referenced by:  brab2a  4927  brab2ga  4951  soirri  5260  sotri  5261  sotri2  5263  sotri3  5264  soirriOLD  5265  sotriOLD  5266  ndmovord  6237  ndmovordi  6238  swoer  6933  brecop2  6998  ecopovsym  7006  ecopovtrn  7007  hartogslem1  7511  nlt1pi  8783  indpi  8784  nqerf  8807  ordpipq  8819  lterpq  8847  ltexnq  8852  ltbtwnnq  8855  ltrnq  8856  prnmadd  8874  genpcd  8883  nqpr  8891  1idpr  8906  ltexprlem4  8916  ltexpri  8920  ltaprlem  8921  prlem936  8924  reclem2pr  8925  reclem3pr  8926  reclem4pr  8927  suplem1pr  8929  suplem2pr  8930  supexpr  8931  recexsrlem  8978  addgt0sr  8979  mulgt0sr  8980  mappsrpr  8983  map2psrpr  8985  supsrlem  8986  supsr  8987  ltresr  9015  dfle2  10740  dflt2  10741  dvdszrcl  12857  letsr  14672  hmphtop  17810  vcex  22059  brtxp2  25726  brpprod3a  25731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884
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