Theorem idaf

Description: The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017)

	Ref	Expression
Hypotheses	idafval.i	|- I = ( IdA ` C )
	idafval.b	|- B = ( Base ` C )
	idafval.c	|- ( ph -> C e. Cat )
	idaf.a	|- A = ( Arrow ` C )
Assertion	idaf	|- ( ph -> I : B --> A )

Proof

Step	Hyp	Ref	Expression
1		idafval.i	|- I = ( IdA ` C )
2		idafval.b	|- B = ( Base ` C )
3		idafval.c	|- ( ph -> C e. Cat )
4		idaf.a	|- A = ( Arrow ` C )
5		otex	|- <. x , x , ( ( Id ` C ) ` x ) >. e. _V
6	5	a1i	|- ( ( ph /\ x e. B ) -> <. x , x , ( ( Id ` C ) ` x ) >. e. _V )
7		eqid	|- ( Id ` C ) = ( Id ` C )
8	1 2 3 7	idafval	|- ( ph -> I = ( x e. B |-> <. x , x , ( ( Id ` C ) ` x ) >. ) )
9		eqid	|- ( HomA ` C ) = ( HomA ` C )
10	4 9	homarw	|- ( x ( HomA ` C ) x ) C_ A
11	3	adantr	|- ( ( ph /\ x e. B ) -> C e. Cat )
12		simpr	|- ( ( ph /\ x e. B ) -> x e. B )
13	1 2 11 12 9	idahom	|- ( ( ph /\ x e. B ) -> ( I ` x ) e. ( x ( HomA ` C ) x ) )
14	10 13	sseldi	|- ( ( ph /\ x e. B ) -> ( I ` x ) e. A )
15	6 8 14	fmpt2d	|- ( ph -> I : B --> A )