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Metamath Proof Explorer

Theorem elhoma

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 elhoma 
|- ( ph -> ( Z ( X H Y ) F <-> ( Z = <. X , Y >. /\ F e. ( 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 1 2 3 4 5 6 homaval 
 |-  ( ph -> ( X H Y ) = ( { <. X , Y >. } X. ( X J Y ) ) )
8 7 breqd 
 |-  ( ph -> ( Z ( X H Y ) F <-> Z ( { <. X , Y >. } X. ( X J Y ) ) F ) )
9 brxp 
 |-  ( Z ( { <. X , Y >. } X. ( X J Y ) ) F <-> ( Z e. { <. X , Y >. } /\ F e. ( X J Y ) ) )
10 opex 
 |-  <. X , Y >. e. _V
11 10 elsn2 
 |-  ( Z e. { <. X , Y >. } <-> Z = <. X , Y >. )
12 11 anbi1i 
 |-  ( ( Z e. { <. X , Y >. } /\ F e. ( X J Y ) ) <-> ( Z = <. X , Y >. /\ F e. ( X J Y ) ) )
13 9 12 bitri 
 |-  ( Z ( { <. X , Y >. } X. ( X J Y ) ) F <-> ( Z = <. X , Y >. /\ F e. ( X J Y ) ) )
14 8 13 bitrdi 
 |-  ( ph -> ( Z ( X H Y ) F <-> ( Z = <. X , Y >. /\ F e. ( X J Y ) ) ) )