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Theorem releldm 4822
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 4627 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
2 brrelex2 4628 . 2  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
3 simpr 109 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A R B )
4 breldmg 4793 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  A R B )  ->  A  e.  dom  R )
51, 2, 3, 4syl3anc 1220 1  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 2128   _Vcvv 2712   class class class wbr 3966   dom cdm 4587   Rel wrel 4592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-xp 4593  df-rel 4594  df-dm 4597
This theorem is referenced by:  releldmb  4824  releldmi  4826  funeu  5196  fnbr  5273  relelfvdm  5501  funbrfv2b  5514  funfvbrb  5581  ercl  6492  dvidlemap  13102  dvmulxxbr  13108  dviaddf  13111  dvimulf  13112  dvcoapbr  13113
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