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 ) ) |
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 ) ) |