Theorem breldmd 5847

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 5844	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1373	1 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2111   class class class wbr 5086  dom cdm 5611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-dm 5621

This theorem is referenced by:  fvelimad  6884  climresdm  45888  xlimliminflimsup  45900