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Metamath Proof Explorer

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