Theorem breldm 4962

Description: Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)

Hypotheses

Ref	Expression
opeldm.1	|- A e. _V
opeldm.2	|- B e. _V

Assertion

Ref	Expression
breldm	|- ( A R B -> A e. dom R )

Proof of Theorem breldm

Step	Hyp	Ref	Expression
1		df-br 4112	. 2 |- ( A R B <-> <. A , B >. e. R )
2		opeldm.1	. . 3 |- A e. _V
3		opeldm.2	. . 3 |- B e. _V
4	2, 3	opeldm 4961	. 2 |- ( <. A , B >. e. R -> A e. dom R )
5	1, 4	sylbi 121	1 |- ( A R B -> A e. dom R )

Syntax hints:   -> wi 4   e. wcel 2205   _Vcvv 2815   <.cop 3694   class class class wbr 4111   dom cdm 4751

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-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-dm 4761

This theorem is referenced by:  exse2  5138  funcnv3  5420  dff13  5943  reldmtpos  6486  rntpos  6490  dftpos4  6496  tpostpos  6497  iserd  6795  ntrivcvgap  12238