Theorem pjval

Description: Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	pjfval2.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{ocv}\left(W\right)$$
	pjfval2.p	$$⊢P={proj}_1\left(W\right)$$
	pjfval2.k	$$⊢K=\mathrm{proj}\left(W\right)$$
Assertion	pjval	$$⊢T\in \mathrm{dom}K\to K\left(T\right)=TP\underset{\dot{}}{\perp }\left(T\right)$$

Proof

Step	Hyp	Ref	Expression
1		pjfval2.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{ocv}\left(W\right)$$
2		pjfval2.p	$$⊢P={proj}_1\left(W\right)$$
3		pjfval2.k	$$⊢K=\mathrm{proj}\left(W\right)$$
4		id	$$⊢x=T\to x=T$$
5		fveq2	$$⊢x=T\to \underset{\dot{}}{\perp }\left(x\right)=\underset{\dot{}}{\perp }\left(T\right)$$
6	4 5	oveq12d	$$⊢x=T\to xP\underset{\dot{}}{\perp }\left(x\right)=TP\underset{\dot{}}{\perp }\left(T\right)$$
7	1 2 3	pjfval2	$$⊢K=\left(x\in \mathrm{dom}K⟼xP\underset{\dot{}}{\perp }\left(x\right)\right)$$
8		ovex	$$⊢TP\underset{\dot{}}{\perp }\left(T\right)\in \mathrm{V}$$
9	6 7 8	fvmpt	$$⊢T\in \mathrm{dom}K\to K\left(T\right)=TP\underset{\dot{}}{\perp }\left(T\right)$$