Theorem nffo

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

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

Proof

Step	Hyp	Ref	Expression
1		nffo.1	⊢ Ⅎ 𝑥 𝐹
2		nffo.2	⊢ Ⅎ 𝑥 𝐴
3		nffo.3	⊢ Ⅎ 𝑥 𝐵
4		df-fo	⊢ ( 𝐹 : 𝐴 –onto→ 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	nfeq	⊢ 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 –onto→ 𝐵