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

Theorem releldm 5014
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 4830 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
2 brrelex2 4831 . 2  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
3 simpr 447 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A R B )
4 breldmg 4987 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  A R B )  ->  A  e.  dom  R )
51, 2, 3, 4syl3anc 1183 1  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1715   _Vcvv 2873   class class class wbr 4125   dom cdm 4792   Rel wrel 4797
This theorem is referenced by:  releldmb  5016  releldmi  5018  sofld  5224  funeu  5381  fnbr  5451  funbrfv2b  5674  funfvbrb  5745  ercl  6813  inviso1  13878  setciso  14133  lmle  18942  dvidlem  19480  dvmulbr  19503  dvcobr  19510  ulmcau  19989  ulmdvlem3  19996  uhgraun  21024  umgraun  21041  uslgraun  21072  heibor1lem  26039  rrncmslem  26062  funbrafv  27529  funbrafv2b  27530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-xp 4798  df-rel 4799  df-dm 4802
  Copyright terms: Public domain W3C validator