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Metamath Proof Explorer

Theorem breldmd

Description: Membership of first of a binary relation in a domain. (Contributed by Glauco Siliprandi, 23-Apr-2023)

Ref Expression
Hypotheses breldmd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
breldmd.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
breldmd.3 ⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
Assertion breldmd ( 𝜑 → 𝐴 ∈ dom 𝑅 )

Proof

Step Hyp Ref Expression
1 breldmd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
2 breldmd.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
3 breldmd.3 ⊢ ( 𝜑 → 𝐴 𝑅 𝐵 )
4 breldmg ⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 𝑅 𝐵 ) → 𝐴 ∈ dom 𝑅 )
5 1 2 3 4 syl3anc ⊢ ( 𝜑 → 𝐴 ∈ dom 𝑅 )