Theorem releldm 3344

Description: The first argument of a binary relation belongs to its domain.

Assertion

Ref	Expression
releldm	|- ((Rel R /\ ARB) -> A e. dom R)

Proof of Theorem releldm

Step	Hyp	Ref	Expression
1		brrelex 3205	. 2 |- ((Rel R /\ ARB) -> A e. V)
2		breldmg 3314	. 2 |- ((A e. V /\ ARB) -> A e. dom R)
3	1, 2	sylancom 475	1 |- ((Rel R /\ ARB) -> A e. dom R)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  Vcvv 1809   class class class wbr 2617  dom cdm 3168  Rel wrel 3173

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-11 967  ax-12 968  ax-13 969  ax-14 970  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  ax-sep 2701  ax-pow 2740  ax-pr 2777

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2618  df-opab 2665  df-xp 3182  df-rel 3183  df-dm 3186

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