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Theorem brel 4888
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 4438 . 2  |-  ( A R B  ->  A
( C  X.  D
) B )
3 brxp 4870 . 2  |-  ( A ( C  X.  D
) B  <->  ( A  e.  C  /\  B  e.  D ) )
42, 3sylib 201 1  |-  ( A R B  ->  ( A  e.  C  /\  B  e.  D )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    e. wcel 1904    C_ wss 3390   class class class wbr 4395    X. cxp 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845
This theorem is referenced by:  brab2a  4889  brab2ga  4915  soirri  5232  sotri  5233  sotri2  5235  sotri3  5236  ndmovord  6478  ndmovordi  6479  swoer  7409  brecop2  7475  ecopovsym  7483  ecopovtrn  7484  hartogslem1  8075  nlt1pi  9349  indpi  9350  nqerf  9373  ordpipq  9385  lterpq  9413  ltexnq  9418  ltbtwnnq  9421  ltrnq  9422  prnmadd  9440  genpcd  9449  nqpr  9457  1idpr  9472  ltexprlem4  9482  ltexpri  9486  ltaprlem  9487  prlem936  9490  reclem2pr  9491  reclem3pr  9492  reclem4pr  9493  suplem1pr  9495  suplem2pr  9496  supexpr  9497  recexsrlem  9545  addgt0sr  9546  mulgt0sr  9547  mappsrpr  9550  map2psrpr  9552  supsrlem  9553  supsr  9554  ltresr  9582  dfle2  11469  dflt2  11470  dvdszrcl  14387  letsr  16551  hmphtop  20870  vcex  26280  brtxp2  30719  brpprod3a  30724
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