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Metamath Proof Explorer

Theorem fnmpti

Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004) (Revised by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Hypotheses	fnmpti.1	⊢ 𝐵 ∈ V
	fnmpti.2	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
Assertion	fnmpti	⊢ 𝐹 Fn 𝐴

Proof

Step	Hyp	Ref	Expression
1		fnmpti.1	⊢ 𝐵 ∈ V
2		fnmpti.2	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
3	1	rgenw	⊢ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V
4	2	mptfng	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴 )
5	3 4	mpbi	⊢ 𝐹 Fn 𝐴