Theorem nff1o

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

	Ref	Expression
Hypotheses	nff1o.1	⊢ Ⅎ 𝑥 𝐹
	nff1o.2	⊢ Ⅎ 𝑥 𝐴
	nff1o.3	⊢ Ⅎ 𝑥 𝐵
Assertion	nff1o	⊢ Ⅎ 𝑥 𝐹 : 𝐴 –1-1-onto→ 𝐵

Proof

Step	Hyp	Ref	Expression
1		nff1o.1	⊢ Ⅎ 𝑥 𝐹
2		nff1o.2	⊢ Ⅎ 𝑥 𝐴
3		nff1o.3	⊢ Ⅎ 𝑥 𝐵
4		df-f1o	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐹 : 𝐴 –onto→ 𝐵 ) )
5	1 2 3	nff1	⊢ Ⅎ 𝑥 𝐹 : 𝐴 –1-1→ 𝐵
6	1 2 3	nffo	⊢ Ⅎ 𝑥 𝐹 : 𝐴 –onto→ 𝐵
7	5 6	nfan	⊢ Ⅎ 𝑥 ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐹 : 𝐴 –onto→ 𝐵 )
8	4 7	nfxfr	⊢ Ⅎ 𝑥 𝐹 : 𝐴 –1-1-onto→ 𝐵