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Theorem releldm 5065
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 4879 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
2 brrelex2 4880 . 2  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
3 simpr 448 . 2  |-  ( ( Rel  R  /\  A R B )  ->  A R B )
4 breldmg 5038 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  A R B )  ->  A  e.  dom  R )
51, 2, 3, 4syl3anc 1184 1  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  dom  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   _Vcvv 2920   class class class wbr 4176   dom cdm 4841   Rel wrel 4846
This theorem is referenced by:  releldmb  5067  releldmi  5069  sofld  5281  funeu  5440  fnbr  5510  funbrfv2b  5734  funfvbrb  5806  ercl  6879  inviso1  13950  setciso  14205  lmle  19211  dvidlem  19759  dvmulbr  19782  dvcobr  19789  ulmcau  20268  ulmdvlem3  20275  uhgraun  21303  umgraun  21320  metideq  24245  heibor1lem  26412  rrncmslem  26435  funbrafv  27893  funbrafv2b  27894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-xp 4847  df-rel 4848  df-dm 4851
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