Theorem nff1

Description: Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004)

	Ref	Expression
Hypotheses	nff1.1	⊢ Ⅎ 𝑥 𝐹
	nff1.2	⊢ Ⅎ 𝑥 𝐴
	nff1.3	⊢ Ⅎ 𝑥 𝐵
Assertion	nff1	⊢ Ⅎ 𝑥 𝐹 : 𝐴 –1-1→ 𝐵

Proof

Step	Hyp	Ref	Expression
1		nff1.1	⊢ Ⅎ 𝑥 𝐹
2		nff1.2	⊢ Ⅎ 𝑥 𝐴
3		nff1.3	⊢ Ⅎ 𝑥 𝐵
4		df-f1	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Fun ◡ 𝐹 ) )
5	1 2 3	nff	⊢ Ⅎ 𝑥 𝐹 : 𝐴 ⟶ 𝐵
6	1	nfcnv	⊢ Ⅎ 𝑥 ◡ 𝐹
7	6	nffun	⊢ Ⅎ 𝑥 Fun ◡ 𝐹
8	5 7	nfan	⊢ Ⅎ 𝑥 ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Fun ◡ 𝐹 )
9	4 8	nfxfr	⊢ Ⅎ 𝑥 𝐹 : 𝐴 –1-1→ 𝐵