Theorem bdayrn

Description: The birthday function's range is On . (Contributed by Scott Fenton, 14-Jun-2011)

	Ref	Expression
Assertion	bdayrn	$$⊢\mathrm{ran}\mathrm{bday}=\mathrm{On}$$

Proof

Step	Hyp	Ref	Expression
1		bdayfo	$$⊢\mathrm{bday}:\mathrm{No}\underset{onto}{⟶}\mathrm{On}$$
2		forn	$$⊢\mathrm{bday}:\mathrm{No}\underset{onto}{⟶}\mathrm{On}\to \mathrm{ran}\mathrm{bday}=\mathrm{On}$$
3	1 2	ax-mp	$$⊢\mathrm{ran}\mathrm{bday}=\mathrm{On}$$