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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