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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 -1
5 1 2 3 nff ⊢ Ⅎ x F : A ⟶ B
6 1 nfcnv ⊢ Ⅎ _ x F -1
7 6 nffun ⊢ Ⅎ x Fun ⁡ F -1
8 5 7 nfan ⊢ Ⅎ x F : A ⟶ B ∧ Fun ⁡ F -1
9 4 8 nfxfr ⊢ Ⅎ x F : A ⟶ 1-1 B