Theorem breldm 3315

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 2620	. 2 |- (ARB <-> <.A, B>. e. R)
2		breldm.1	. . 3 |- A e. V
3	2	opeldm 3314	. 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 958  Vcvv 1811  <.cop 2411   class class class wbr 2619  dom cdm 3170

This theorem is referenced by:  breldmg 3316  asymref 3439  asymref2 3440  funcnv3 3558  f1fv 3874  cbvfo 3885  ereldm 4285  psdmrn 8648  bra11 10041  dmhmpha 10534

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-nul 2281  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-dm 3188

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