Theorem relssdv

Description: Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004)

	Ref	Expression
Hypotheses	relssdv.1	⊢ ( 𝜑 → Rel 𝐴 )
	relssdv.2	⊢ ( 𝜑 → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
Assertion	relssdv	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		relssdv.1	⊢ ( 𝜑 → Rel 𝐴 )
2		relssdv.2	⊢ ( 𝜑 → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
3	2	alrimivv	⊢ ( 𝜑 → ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
4		ssrel	⊢ ( Rel 𝐴 → ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) ) )
5	1 4	syl	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) ) )
6	3 5	mpbird	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )