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Metamath Proof Explorer

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