Theorem releldm 4902

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

Step	Hyp	Ref	Expression
1		brrelex 4704	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
2		brrelex2 4705	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3		simpr 110	. 2 |- ( ( Rel R /\ A R B ) -> A R B )
4		breldmg 4873	. 2 |- ( ( A e. _V /\ B e. _V /\ A R B ) -> A e. dom R )
5	1, 2, 3, 4	syl3anc 1249	1 |- ( ( Rel R /\ A R B ) -> A e. dom R )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2167   _Vcvv 2763   class class class wbr 4034   dom cdm 4664   Rel wrel 4669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-dm 4674

This theorem is referenced by:  releldmb  4904  releldmi  4906  funeu  5284  fnbr  5363  relelfvdm  5593  funbrfv2b  5608  funfvbrb  5678  ercl  6612  dvidlemap  15011  dvidrelem  15012  dvidsslem  15013  dvmulxxbr  15022  dviaddf  15025  dvimulf  15026  dvcoapbr  15027