Theorem releldm 5358

Description: The first argument of a binary relation belongs to its domain. Note that A R B does not imply Rel R: see for example nrelv 5244 and brv 4941. (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 5156	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
2		brrelex2 5157	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3		simpr 477	. 2 |- ( ( Rel R /\ A R B ) -> A R B )
4		breldmg 5330	. 2 |- ( ( A e. _V /\ B e. _V /\ A R B ) -> A e. dom R )
5	1, 2, 3, 4	syl3anc 1326	1 |- ( ( Rel R /\ A R B ) -> A e. dom R )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   dom cdm 5114   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124

This theorem is referenced by:  releldmb  5360  releldmi  5362  sofld  5581  funeu  5913  fnbr  5993  funbrfv2b  6240  funfvbrb  6330  ercl  7753  inviso1  16426  setciso  16741  lmle  23099  dvidlem  23679  dvmulbr  23702  dvcobr  23709  ulmcau  24149  ulmdvlem3  24156  metideq  29936  heibor1lem  33608  rrncmslem  33631  ntrclsiex  38351  ntrneiiex  38374  binomcxplemnn0  38548  binomcxplemnotnn0  38555  sumnnodd  39862  climlimsup  39992  climlimsupcex  40001  climliminflimsupd  40033  liminflimsupclim  40039  ioodvbdlimc1lem2  40147  ioodvbdlimc2lem  40149  funbrafv  41238  funbrafv2b  41239  rngciso  41982  rngcisoALTV  41994  ringciso  42033  ringcisoALTV  42057