Theorem fnhomeqhomf

Description: If the Hom-set operation is a function it is equal to the corresponding functionalized Hom-set operation. (Contributed by AV, 1-Mar-2020)

	Ref	Expression
Hypotheses	homffval.f	|- F = ( Homf ` C )
	homffval.b	|- B = ( Base ` C )
	homffval.h	|- H = ( Hom ` C )
Assertion	fnhomeqhomf	|- ( H Fn ( B X. B ) -> F = H )

Proof

Step	Hyp	Ref	Expression
1		homffval.f	|- F = ( Homf ` C )
2		homffval.b	|- B = ( Base ` C )
3		homffval.h	|- H = ( Hom ` C )
4		fnov	|- ( H Fn ( B X. B ) <-> H = ( x e. B , y e. B |-> ( x H y ) ) )
5	1 2 3	homffval	|- F = ( x e. B , y e. B |-> ( x H y ) )
6		eqeq2	|- ( H = ( x e. B , y e. B |-> ( x H y ) ) -> ( F = H <-> F = ( x e. B , y e. B |-> ( x H y ) ) ) )
7	5 6	mpbiri	|- ( H = ( x e. B , y e. B |-> ( x H y ) ) -> F = H )
8	4 7	sylbi	|- ( H Fn ( B X. B ) -> F = H )