Metamath Proof Explorer

Theorem fucterm

Description: The category of functors to a terminal category is terminal. (Contributed by Zhi Wang, 17-Nov-2025)

Ref Expression
Hypotheses funcsn.q 
|- Q = ( C FuncCat D )
fucterm.c 
|- ( ph -> C e. Cat )
fucterm.d 
|- ( ph -> D e. TermCat )
Assertion fucterm 
|- ( ph -> Q e. TermCat )

Proof

Step Hyp Ref Expression
1 funcsn.q 
 |-  Q = ( C FuncCat D )
2 fucterm.c 
 |-  ( ph -> C e. Cat )
3 fucterm.d 
 |-  ( ph -> D e. TermCat )
4 opex 
 |-  <. ( ( Base ` C ) X. ( Base ` D ) ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x ( Hom ` C ) y ) X. ( ( ( ( Base ` C ) X. ( Base ` D ) ) ` x ) ( Hom ` D ) ( ( ( Base ` C ) X. ( Base ` D ) ) ` y ) ) ) ) >. e. _V
5 4 a1i 
 |-  ( ph -> <. ( ( Base ` C ) X. ( Base ` D ) ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x ( Hom ` C ) y ) X. ( ( ( ( Base ` C ) X. ( Base ` D ) ) ` x ) ( Hom ` D ) ( ( ( Base ` C ) X. ( Base ` D ) ) ` y ) ) ) ) >. e. _V )
6 eqid 
 |-  ( Base ` C ) = ( Base ` C )
7 eqid 
 |-  ( Base ` D ) = ( Base ` D )
8 eqid 
 |-  ( Hom ` C ) = ( Hom ` C )
9 eqid 
 |-  ( Hom ` D ) = ( Hom ` D )
10 eqid 
 |-  ( ( Base ` C ) X. ( Base ` D ) ) = ( ( Base ` C ) X. ( Base ` D ) )
11 eqid 
 |-  ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x ( Hom ` C ) y ) X. ( ( ( ( Base ` C ) X. ( Base ` D ) ) ` x ) ( Hom ` D ) ( ( ( Base ` C ) X. ( Base ` D ) ) ` y ) ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x ( Hom ` C ) y ) X. ( ( ( ( Base ` C ) X. ( Base ` D ) ) ` x ) ( Hom ` D ) ( ( ( Base ` C ) X. ( Base ` D ) ) ` y ) ) ) )
12 2 3 6 7 8 9 10 11 functermc2 
 |-  ( ph -> ( C Func D ) = { <. ( ( Base ` C ) X. ( Base ` D ) ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x ( Hom ` C ) y ) X. ( ( ( ( Base ` C ) X. ( Base ` D ) ) ` x ) ( Hom ` D ) ( ( ( Base ` C ) X. ( Base ` D ) ) ` y ) ) ) ) >. } )
13 3 termcthind 
 |-  ( ph -> D e. ThinCat )
14 1 5 12 13 funcsn 
 |-  ( ph -> Q e. TermCat )