Theorem fpm

Description: A total function is a partial function. (Contributed by NM, 15-Nov-2007) (Revised by Mario Carneiro, 31-Dec-2013)

	Ref	Expression
Hypotheses	elmap.1	$$⊢A\in \mathrm{V}$$
	elmap.2	$$⊢B\in \mathrm{V}$$
Assertion	fpm	$$⊢F:A⟶B\to F\in \left(B{\uparrow }_{𝑝𝑚}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		elmap.1	$$⊢A\in \mathrm{V}$$
2		elmap.2	$$⊢B\in \mathrm{V}$$
3		fpmg	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\wedge F:A⟶B\right)\to F\in \left(B{\uparrow }_{𝑝𝑚}A\right)$$
4	1 2 3	mp3an12	$$⊢F:A⟶B\to F\in \left(B{\uparrow }_{𝑝𝑚}A\right)$$