MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  releldm Structured version   Unicode version

Theorem releldm 5131
Description: The first argument of a binary relation belongs to its domain. (Contributed by NM, 2-Jul-2008.)
Assertion
Ref Expression
releldm  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  dom  R )

Proof of Theorem releldm
StepHypRef Expression
1 brrelex 4945 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
2 brrelex2 4946 . 2  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
3 simpr 449 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A R B )
4 breldmg 5104 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  A R B )  ->  A  e.  dom  R )
51, 2, 3, 4syl3anc 1185 1  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1727   _Vcvv 2962   class class class wbr 4237   dom cdm 4907   Rel wrel 4912
This theorem is referenced by:  releldmb  5133  releldmi  5135  sofld  5347  funeu  5506  fnbr  5576  funbrfv2b  5800  funfvbrb  5872  ercl  6945  inviso1  14022  setciso  14277  lmle  19285  dvidlem  19833  dvmulbr  19856  dvcobr  19863  ulmcau  20342  ulmdvlem3  20349  uhgraun  21377  umgraun  21394  metideq  24319  heibor1lem  26556  rrncmslem  26579  funbrafv  28036  funbrafv2b  28037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-xp 4913  df-rel 4914  df-dm 4917
  Copyright terms: Public domain W3C validator