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Theorem brelg 26093
Description: Two things in a binary relation belong to the relation's domain. (Contributed by Thierry Arnoux, 29-Aug-2017.)
Assertion
Ref Expression
brelg  |-  ( ( R  C_  ( C  X.  D )  /\  A R B )  ->  ( A  e.  C  /\  B  e.  D )
)

Proof of Theorem brelg
StepHypRef Expression
1 id 22 . . . 4  |-  ( R 
C_  ( C  X.  D )  ->  R  C_  ( C  X.  D
) )
21ssbrd 4442 . . 3  |-  ( R 
C_  ( C  X.  D )  ->  ( A R B  ->  A
( C  X.  D
) B ) )
32imp 429 . 2  |-  ( ( R  C_  ( C  X.  D )  /\  A R B )  ->  A
( C  X.  D
) B )
4 brxp 4979 . 2  |-  ( A ( C  X.  D
) B  <->  ( A  e.  C  /\  B  e.  D ) )
53, 4sylib 196 1  |-  ( ( R  C_  ( C  X.  D )  /\  A R B )  ->  ( A  e.  C  /\  B  e.  D )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758    C_ wss 3437   class class class wbr 4401    X. cxp 4947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-xp 4955
This theorem is referenced by:  fpwrelmap  26185
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