Theorem fnom

Description: Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995) (Revised by Mario Carneiro, 8-Sep-2013)

	Ref	Expression
Assertion	fnom	$$⊢{\cdot }_{𝑜}Fn\left(\mathrm{On}\times \mathrm{On}\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-omul	$$⊢{\cdot }_{𝑜}=\left(x\in \mathrm{On},y\in \mathrm{On}⟼\mathrm{rec}\left(\left(z\in \mathrm{V}⟼z{+}_{𝑜}x\right),\varnothing \right)\left(y\right)\right)$$
2		fvex	$$⊢\mathrm{rec}\left(\left(z\in \mathrm{V}⟼z{+}_{𝑜}x\right),\varnothing \right)\left(y\right)\in \mathrm{V}$$
3	1 2	fnmpoi	$$⊢{\cdot }_{𝑜}Fn\left(\mathrm{On}\times \mathrm{On}\right)$$