Theorem fpmg

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

	Ref	Expression
Assertion	fpmg	$$⊢\left(A\in V\wedge B\in W\wedge F:A⟶B\right)\to F\in \left(B{\uparrow }_{𝑝𝑚}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		ssid	$$⊢A\subseteq A$$
2		elpm2r	$$⊢\left(\left(B\in W\wedge A\in V\right)\wedge \left(F:A⟶B\wedge A\subseteq A\right)\right)\to F\in \left(B{\uparrow }_{𝑝𝑚}A\right)$$
3	1 2	mpanr2	$$⊢\left(\left(B\in W\wedge A\in V\right)\wedge F:A⟶B\right)\to F\in \left(B{\uparrow }_{𝑝𝑚}A\right)$$
4	3	3impa	$$⊢\left(B\in W\wedge A\in V\wedge F:A⟶B\right)\to F\in \left(B{\uparrow }_{𝑝𝑚}A\right)$$
5	4	3com12	$$⊢\left(A\in V\wedge B\in W\wedge F:A⟶B\right)\to F\in \left(B{\uparrow }_{𝑝𝑚}A\right)$$