Theorem releldm 4973

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 4772	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
2		brrelex2 4773	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3		simpr 110	. 2 |- ( ( Rel R /\ A R B ) -> A R B )
4		breldmg 4943	. 2 |- ( ( A e. _V /\ B e. _V /\ A R B ) -> A e. dom R )
5	1, 2, 3, 4	syl3anc 1274	1 |- ( ( Rel R /\ A R B ) -> A e. dom R )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   _Vcvv 2803   class class class wbr 4093   dom cdm 4731   Rel wrel 4736

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-dm 4741

This theorem is referenced by:  releldmb  4975  releldmi  4977  funeu  5358  fnbr  5441  relelfvdm  5680  funbrfv2b  5699  funfvbrb  5769  ercl  6756  dvidlemap  15502  dvidrelem  15503  dvidsslem  15504  dvmulxxbr  15513  dviaddf  15516  dvimulf  15517  dvcoapbr  15518