Metamath Proof Explorer

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 syl5eq 
 |-  ( ph -> ( X H Y ) = ( { <. X , Y >. } X. ( X J Y ) ) )