Metamath Proof Explorer


Theorem nofnbday

Description: A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011)

Ref Expression
Assertion nofnbday ( 𝐴 No 𝐴 Fn ( bday 𝐴 ) )

Proof

Step Hyp Ref Expression
1 nofun ( 𝐴 No → Fun 𝐴 )
2 bdayval ( 𝐴 No → ( bday 𝐴 ) = dom 𝐴 )
3 2 eqcomd ( 𝐴 No → dom 𝐴 = ( bday 𝐴 ) )
4 df-fn ( 𝐴 Fn ( bday 𝐴 ) ↔ ( Fun 𝐴 ∧ dom 𝐴 = ( bday 𝐴 ) ) )
5 1 3 4 sylanbrc ( 𝐴 No 𝐴 Fn ( bday 𝐴 ) )