Theorem fnopab

Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996)

	Ref	Expression
Hypotheses	fnopab.1	⊢ ( 𝑥 ∈ 𝐴 → ∃! 𝑦 𝜑 )
	fnopab.2	⊢ 𝐹 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
Assertion	fnopab	⊢ 𝐹 Fn 𝐴

Proof

Step	Hyp	Ref	Expression
1		fnopab.1	⊢ ( 𝑥 ∈ 𝐴 → ∃! 𝑦 𝜑 )
2		fnopab.2	⊢ 𝐹 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
3	1	rgen	⊢ ∀ 𝑥 ∈ 𝐴 ∃! 𝑦 𝜑
4	2	fnopabg	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃! 𝑦 𝜑 ↔ 𝐹 Fn 𝐴 )
5	3 4	mpbi	⊢ 𝐹 Fn 𝐴