Theorem breldmd 5884

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

Hypotheses

Ref	Expression
breldmd.1	⊢ (𝜑 → 𝐴 ∈ 𝐶)
breldmd.2	⊢ (𝜑 → 𝐵 ∈ 𝐷)
breldmd.3	⊢ (𝜑 → 𝐴𝑅𝐵)

Assertion

Ref	Expression
breldmd	⊢ (𝜑 → 𝐴 ∈ dom 𝑅)

Proof of Theorem breldmd

Step	Hyp	Ref	Expression
1		breldmd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
2		breldmd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
3		breldmd.3	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
4		breldmg 5881	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1389	1 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2141   class class class wbr 5097  dom cdm 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-dm 5653

This theorem is referenced by:  fvelimad  6929  climresdm  46385  xlimliminflimsup  46397