Theorem breldmd 5905

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 5902	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
5	1, 2, 3, 4	syl3anc 1368	1 ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2098   class class class wbr 5141  dom cdm 5669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-dm 5679

This theorem is referenced by:  fvelimad  6952  climresdm  45119  xlimliminflimsup  45131