Theorem releldm 4587

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 4400	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
2		brrelex2 4401	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3		simpr 108	. 2 |- ( ( Rel R /\ A R B ) -> A R B )
4		breldmg 4559	. 2 |- ( ( A e. _V /\ B e. _V /\ A R B ) -> A e. dom R )
5	1, 2, 3, 4	syl3anc 1169	1 |- ( ( Rel R /\ A R B ) -> A e. dom R )

Syntax hints:   -> wi 4   /\ wa 102   e. wcel 1433   _Vcvv 2601   class class class wbr 3785   dom cdm 4363   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-dm 4373

This theorem is referenced by:  releldmb  4589  releldmi  4591  funeu  4946  fnbr  5021  relelfvdm  5226  funbrfv2b  5239  funfvbrb  5301  ercl  6140