Theorem breldmd 5810

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 5807	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1369	1 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2108   class class class wbr 5070  dom cdm 5580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-dm 5590

This theorem is referenced by:  fvelimad  6818  climresdm  43281  xlimliminflimsup  43293