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 𝑥 𝐹
nff.2 𝑥 𝐴
nff.3 𝑥 𝐵
Assertion nff 𝑥 𝐹 : 𝐴𝐵

Proof

Step Hyp Ref Expression
1 nff.1 𝑥 𝐹
2 nff.2 𝑥 𝐴
3 nff.3 𝑥 𝐵
4 df-f ( 𝐹 : 𝐴𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹𝐵 ) )
5 1 2 nffn 𝑥 𝐹 Fn 𝐴
6 1 nfrn 𝑥 ran 𝐹
7 6 3 nfss 𝑥 ran 𝐹𝐵
8 5 7 nfan 𝑥 ( 𝐹 Fn 𝐴 ∧ ran 𝐹𝐵 )
9 4 8 nfxfr 𝑥 𝐹 : 𝐴𝐵