Theorem diagval

Description: Define the diagonal functor, which is the functor C --> ( D Func C ) whose object part is x e. C |-> ( y e. D |-> x ) . We can define this equationally as the currying of the first projection functor, and by expressing it this way we get a quick proof of functoriality. (Contributed by Mario Carneiro, 6-Jan-2017) (Revised by Mario Carneiro, 15-Jan-2017)

	Ref	Expression
Hypotheses	diagval.l	|- L = ( C DiagFunc D )
	diagval.c	|- ( ph -> C e. Cat )
	diagval.d	|- ( ph -> D e. Cat )
Assertion	diagval	|- ( ph -> L = ( <. C , D >. curryF ( C 1stF D ) ) )

Proof

Step	Hyp	Ref	Expression
1		diagval.l	|- L = ( C DiagFunc D )
2		diagval.c	|- ( ph -> C e. Cat )
3		diagval.d	|- ( ph -> D e. Cat )
4		df-diag	|- DiagFunc = ( c e. Cat , d e. Cat |-> ( <. c , d >. curryF ( c 1stF d ) ) )
5	4	a1i	|- ( ph -> DiagFunc = ( c e. Cat , d e. Cat |-> ( <. c , d >. curryF ( c 1stF d ) ) ) )
6		simprl	|- ( ( ph /\ ( c = C /\ d = D ) ) -> c = C )
7		simprr	|- ( ( ph /\ ( c = C /\ d = D ) ) -> d = D )
8	6 7	opeq12d	|- ( ( ph /\ ( c = C /\ d = D ) ) -> <. c , d >. = <. C , D >. )
9	6 7	oveq12d	|- ( ( ph /\ ( c = C /\ d = D ) ) -> ( c 1stF d ) = ( C 1stF D ) )
10	8 9	oveq12d	|- ( ( ph /\ ( c = C /\ d = D ) ) -> ( <. c , d >. curryF ( c 1stF d ) ) = ( <. C , D >. curryF ( C 1stF D ) ) )
11		ovexd	|- ( ph -> ( <. C , D >. curryF ( C 1stF D ) ) e. _V )
12	5 10 2 3 11	ovmpod	|- ( ph -> ( C DiagFunc D ) = ( <. C , D >. curryF ( C 1stF D ) ) )
13	1 12	eqtrid	|- ( ph -> L = ( <. C , D >. curryF ( C 1stF D ) ) )