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Theorem brelg 27780
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 4433 . . 3  |-  ( R 
C_  ( C  X.  D )  ->  ( A R B  ->  A
( C  X.  D
) B ) )
32imp 427 . 2  |-  ( ( R  C_  ( C  X.  D )  /\  A R B )  ->  A
( C  X.  D
) B )
4 brxp 4971 . 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 367    e. wcel 1840    C_ wss 3411   class class class wbr 4392    X. cxp 4938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-br 4393  df-opab 4451  df-xp 4946
This theorem is referenced by:  fpwrelmap  27884
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