Theorem dmopabss

Description: Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004)

	Ref	Expression
Assertion	dmopabss	⊢ dom { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ⊆ 𝐴

Proof

Step	Hyp	Ref	Expression
1		dmopab	⊢ dom { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } = { 𝑥 ∣ ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
2		19.42v	⊢ ( ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝜑 ) )
3	2	abbii	⊢ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝜑 ) }
4		ssab2	⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝜑 ) } ⊆ 𝐴
5	3 4	eqsstri	⊢ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ⊆ 𝐴
6	1 5	eqsstri	⊢ dom { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ⊆ 𝐴