Theorem breldm 3321

Description: Membership of first of a binary relation in a domain.

Hypothesis

Ref	Expression
breldm.1	|- A e. V

Assertion

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

Proof of Theorem breldm

Step	Hyp	Ref	Expression
1		df-br 2625	. 2 |- (ARB <-> <.A, B>. e. R)
2		breldm.1	. . 3 |- A e. V
3	2	opeldm 3320	. 2 |- (<.A, B>. e. R -> A e. dom R)
4	1, 3	sylbi 199	1 |- (ARB -> A e. dom R)

Syntax hints:   -> wi 3   e. wcel 960  Vcvv 1814  <.cop 2415   class class class wbr 2624  dom cdm 3176

This theorem is referenced by:  breldmg 3322  asymref 3445  asymref2 3446  funcnv3 3564  f1fv 3880  cbvfo 3891  ereldm 4291  psdmrn 8644  bra11 10036  dmhmpha 10520

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-nul 2284  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-dm 3194

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