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Theorem brel 5040
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 4482 . 2  |-  ( A R B  ->  A
( C  X.  D
) B )
3 brxp 5022 . 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 1762    C_ wss 3469   class class class wbr 4440    X. cxp 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998
This theorem is referenced by:  brab2a  5041  brab2ga  5066  soirri  5384  sotri  5385  sotri2  5387  sotri3  5388  soirriOLD  5389  sotriOLD  5390  ndmovord  6440  ndmovordi  6441  swoer  7329  brecop2  7395  ecopovsym  7403  ecopovtrn  7404  hartogslem1  7956  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  11342  dflt2  11343  dvdszrcl  13841  letsr  15703  hmphtop  20007  vcex  25135  brtxp2  29094  brpprod3a  29099
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