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Theorem brel 4885
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 4332 . 2  |-  ( A R B  ->  A
( C  X.  D
) B )
3 brxp 4868 . 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 369    e. wcel 1756    C_ wss 3326   class class class wbr 4290    X. cxp 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-br 4291  df-opab 4349  df-xp 4844
This theorem is referenced by:  brab2a  4886  brab2ga  4910  soirri  5222  sotri  5223  sotri2  5225  sotri3  5226  soirriOLD  5227  sotriOLD  5228  ndmovord  6251  ndmovordi  6252  swoer  7127  brecop2  7192  ecopovsym  7200  ecopovtrn  7201  hartogslem1  7754  nlt1pi  9073  indpi  9074  nqerf  9097  ordpipq  9109  lterpq  9137  ltexnq  9142  ltbtwnnq  9145  ltrnq  9146  prnmadd  9164  genpcd  9173  nqpr  9181  1idpr  9196  ltexprlem4  9206  ltexpri  9210  ltaprlem  9211  prlem936  9214  reclem2pr  9215  reclem3pr  9216  reclem4pr  9217  suplem1pr  9219  suplem2pr  9220  supexpr  9221  recexsrlem  9268  addgt0sr  9269  mulgt0sr  9270  mappsrpr  9273  map2psrpr  9275  supsrlem  9276  supsr  9277  ltresr  9305  dfle2  11122  dflt2  11123  dvdszrcl  13538  letsr  15395  hmphtop  19349  vcex  23956  brtxp2  27910  brpprod3a  27915
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