Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brelg Structured version   Unicode version

Theorem brelg 27121
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 4481 . . 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 5022 . 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 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:  fpwrelmap  27214
  Copyright terms: Public domain W3C validator