Theorem homaval

Description: Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017)

	Ref	Expression
Hypotheses	homarcl.h	|- H = ( HomA ` C )
	homafval.b	|- B = ( Base ` C )
	homafval.c	|- ( ph -> C e. Cat )
	homaval.j	|- J = ( Hom ` C )
	homaval.x	|- ( ph -> X e. B )
	homaval.y	|- ( ph -> Y e. B )
Assertion	homaval	|- ( ph -> ( X H Y ) = ( { <. X , Y >. } X. ( X J Y ) ) )

Proof

Step	Hyp	Ref	Expression
1		homarcl.h	|- H = ( HomA ` C )
2		homafval.b	|- B = ( Base ` C )
3		homafval.c	|- ( ph -> C e. Cat )
4		homaval.j	|- J = ( Hom ` C )
5		homaval.x	|- ( ph -> X e. B )
6		homaval.y	|- ( ph -> Y e. B )
7		df-ov	|- ( X H Y ) = ( H ` <. X , Y >. )
8	1 2 3 4	homafval	|- ( ph -> H = ( z e. ( B X. B ) |-> ( { z } X. ( J ` z ) ) ) )
9		simpr	|- ( ( ph /\ z = <. X , Y >. ) -> z = <. X , Y >. )
10	9	sneqd	|- ( ( ph /\ z = <. X , Y >. ) -> { z } = { <. X , Y >. } )
11	9	fveq2d	|- ( ( ph /\ z = <. X , Y >. ) -> ( J ` z ) = ( J ` <. X , Y >. ) )
12		df-ov	|- ( X J Y ) = ( J ` <. X , Y >. )
13	11 12	eqtr4di	|- ( ( ph /\ z = <. X , Y >. ) -> ( J ` z ) = ( X J Y ) )
14	10 13	xpeq12d	|- ( ( ph /\ z = <. X , Y >. ) -> ( { z } X. ( J ` z ) ) = ( { <. X , Y >. } X. ( X J Y ) ) )
15	5 6	opelxpd	|- ( ph -> <. X , Y >. e. ( B X. B ) )
16		snex	|- { <. X , Y >. } e. _V
17		ovex	|- ( X J Y ) e. _V
18	16 17	xpex	|- ( { <. X , Y >. } X. ( X J Y ) ) e. _V
19	18	a1i	|- ( ph -> ( { <. X , Y >. } X. ( X J Y ) ) e. _V )
20	8 14 15 19	fvmptd	|- ( ph -> ( H ` <. X , Y >. ) = ( { <. X , Y >. } X. ( X J Y ) ) )
21	7 20	eqtrid	|- ( ph -> ( X H Y ) = ( { <. X , Y >. } X. ( X J Y ) ) )