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Theorem releldm 4913
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 4729 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
2 brrelex2 4730 . 2  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
3 simpr 447 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A R B )
4 breldmg 4886 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  A R B )  ->  A  e.  dom  R )
51, 2, 3, 4syl3anc 1182 1  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1686   _Vcvv 2790   class class class wbr 4025   dom cdm 4691   Rel wrel 4696
This theorem is referenced by:  releldmb  4915  releldmi  4917  sofld  5123  funeu  5280  fnbr  5348  funbrfv2b  5569  funfvbrb  5640  ercl  6673  inviso1  13670  setciso  13925  lmle  18729  dvidlem  19267  dvmulbr  19290  dvcobr  19297  ulmcau  19774  ulmdvlem3  19781  umgraun  23881  pre1befi2  25242  heibor1lem  26544  rrncmslem  26567  funbrafv  28031  funbrafv2b  28032  uslgraun  28131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-xp 4697  df-rel 4698  df-dm 4701
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