Theorem nff

Description: Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004) (Revised by Mario Carneiro, 15-Oct-2016)

	Ref	Expression
Hypotheses	nff.1	⊢ Ⅎ 𝑥 𝐹
	nff.2	⊢ Ⅎ 𝑥 𝐴
	nff.3	⊢ Ⅎ 𝑥 𝐵
Assertion	nff	⊢ Ⅎ 𝑥 𝐹 : 𝐴 ⟶ 𝐵

Proof

Step	Hyp	Ref	Expression
1		nff.1	⊢ Ⅎ 𝑥 𝐹
2		nff.2	⊢ Ⅎ 𝑥 𝐴
3		nff.3	⊢ Ⅎ 𝑥 𝐵
4		df-f	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) )
5	1 2	nffn	⊢ Ⅎ 𝑥 𝐹 Fn 𝐴
6	1	nfrn	⊢ Ⅎ 𝑥 ran 𝐹
7	6 3	nfss	⊢ Ⅎ 𝑥 ran 𝐹 ⊆ 𝐵
8	5 7	nfan	⊢ Ⅎ 𝑥 ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 )
9	4 8	nfxfr	⊢ Ⅎ 𝑥 𝐹 : 𝐴 ⟶ 𝐵