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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