Theorem breldm 2535

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 2063	. 2 ⊢ (ARB ↔ ⟨A, B⟩ ∈ R)
2		breldm.1	. . 3 ⊢ A ∈ V
3	2	opeldm 2534	. 2 ⊢ (⟨A, B⟩ ∈ R → A ∈ dom R)
4	1, 3	sylbi 174	1 ⊢ (ARB → A ∈ dom R)

Syntax hints:   → wi 2   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054  dom cdm 2410

This theorem is referenced by:  brelrn 2559  fnbr 2726  f1fv 2916  cbvfo 2923  ereldm 3222

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-dm 2428

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