Theorem releldm 5172

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 4977	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
2		brrelex2 4978	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3		simpr 461	. 2 |- ( ( Rel R /\ A R B ) -> A R B )
4		breldmg 5145	. 2 |- ( ( A e. _V /\ B e. _V /\ A R B ) -> A e. dom R )
5	1, 2, 3, 4	syl3anc 1219	1 |- ( ( Rel R /\ A R B ) -> A e. dom R )

Syntax hints:   -> wi 4   /\ wa 369   e. wcel 1758   _Vcvv 3070   class class class wbr 4392   dom cdm 4940   Rel wrel 4945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-dm 4950

This theorem is referenced by:  releldmb  5174  releldmi  5176  sofld  5386  funeu  5542  fnbr  5613  funbrfv2b  5837  funfvbrb  5917  ercl  7214  inviso1  14808  setciso  15063  lmle  20930  dvidlem  21508  dvmulbr  21531  dvcobr  21538  ulmcau  21978  ulmdvlem3  21985  uhgraun  23382  umgraun  23399  metideq  26456  heibor1lem  28848  rrncmslem  28871  funbrafv  30204  funbrafv2b  30205