Theorem breldm 3331

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

Hypothesis

Ref	Expression
breldm.1	⊢ A ∈ V

Assertion

Ref	Expression
breldm	⊢ (ARB → A ∈ dom R)

Proof of Theorem breldm

Step	Hyp	Ref	Expression
1		df-br 2635	. 2 ⊢ (ARB ↔ ⟨A, B⟩ ∈ R)
2		breldm.1	. . 3 ⊢ A ∈ V
3	2	opeldm 3330	. 2 ⊢ (⟨A, B⟩ ∈ R → A ∈ dom R)
4	1, 3	sylbi 199	1 ⊢ (ARB → A ∈ dom R)

Syntax hints:   → wi 3   ∈ wcel 962  Vcvv 1818  ⟨cop 2423   class class class wbr 2634  dom cdm 3186

This theorem is referenced by:  breldmg 3332  asymref 3455  asymref2 3456  funcnv3 3574  f1fv 3890  cbvfo 3901  ereldm 4301  psdmrn 8673  bra11 10065  domrngref 10499  dmhmpha 10567

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-v 1819  df-dif 2060  df-un 2061  df-nul 2292  df-sn 2424  df-pr 2425  df-op 2428  df-br 2635  df-dm 3204

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