Theorem breldmd 5774

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 5771	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1366	1 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2113   class class class wbr 5059  dom cdm 5548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-br 5060  df-dm 5558

This theorem is referenced by:  fvelimad  6725  climresdm  42206  xlimliminflimsup  42218