Theorem releldm 4914

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 4716	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
2		brrelex2 4717	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3		simpr 110	. 2 |- ( ( Rel R /\ A R B ) -> A R B )
4		breldmg 4885	. 2 |- ( ( A e. _V /\ B e. _V /\ A R B ) -> A e. dom R )
5	1, 2, 3, 4	syl3anc 1250	1 |- ( ( Rel R /\ A R B ) -> A e. dom R )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2176   _Vcvv 2772   class class class wbr 4045   dom cdm 4676   Rel wrel 4681

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-rel 4683  df-dm 4686

This theorem is referenced by:  releldmb  4916  releldmi  4918  funeu  5297  fnbr  5379  relelfvdm  5610  funbrfv2b  5625  funfvbrb  5695  ercl  6633  dvidlemap  15196  dvidrelem  15197  dvidsslem  15198  dvmulxxbr  15207  dviaddf  15210  dvimulf  15211  dvcoapbr  15212