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Metamath Proof Explorer

Theorem nff

Description: Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004) (Revised by Mario Carneiro, 15-Oct-2016)

Ref Expression
Hypotheses nff.1 ⊢ Ⅎ 𝑥 𝐹
nff.2 ⊢ Ⅎ 𝑥 𝐴
nff.3 ⊢ Ⅎ 𝑥 𝐵
Assertion nff Ⅎ 𝑥 𝐹 : 𝐴 ⟶ 𝐵

Proof

Step Hyp Ref Expression
1 nff.1 ⊢ Ⅎ 𝑥 𝐹
2 nff.2 ⊢ Ⅎ 𝑥 𝐴
3 nff.3 ⊢ Ⅎ 𝑥 𝐵
4 df-f ⊢ ( 𝐹 : 𝐴 ⟶ 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 nfss ⊢ 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 ⟶ 𝐵