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Metamath Proof Explorer

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