Theorem dmss

Description: Subset theorem for domain. (Contributed by NM, 11-Aug-1994)

	Ref	Expression
Assertion	dmss	⊢ ( 𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ssel	⊢ ( 𝐴 ⊆ 𝐵 → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
2	1	eximdv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) )
3		vex	⊢ 𝑥 ∈ V
4	3	eldm2	⊢ ( 𝑥 ∈ dom 𝐴 ↔ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 )
5	3	eldm2	⊢ ( 𝑥 ∈ dom 𝐵 ↔ ∃ 𝑦 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 )
6	2 4 5	3imtr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ dom 𝐴 → 𝑥 ∈ dom 𝐵 ) )
7	6	ssrdv	⊢ ( 𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵 )