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Metamath Proof Explorer

Theorem oppctermco

Description: The opposite category of a terminal category has the same base, hom-sets and composition operation as the original category. Note that C = O cannot be proved because C might not even be a function. For example, let C be ( { <. ( Basendx ) , { (/) } >. , <. ( Homndx ) , ( (V X. V ) X. { { (/) } } ) >. } u. { <. ( compndx ) , { (/) } >. , <. ( compndx ) , 2o >. } ) ; it should be a terminal category, but the opposite category is not itself. See the definitions df-oppc and df-sets . (Contributed by Zhi Wang, 16-Oct-2025)

Ref Expression
Hypotheses oppcterm.o 
|- O = ( oppCat ` C )
oppcterm.c 
|- ( ph -> C e. TermCat )
Assertion oppctermco 
|- ( ph -> ( comf ` C ) = ( comf ` O ) )

Proof

Step Hyp Ref Expression
1 oppcterm.o 
 |-  O = ( oppCat ` C )
2 oppcterm.c 
 |-  ( ph -> C e. TermCat )
3 2 termcthind 
 |-  ( ph -> C e. ThinCat )
4 1 2 oppctermhom 
 |-  ( ph -> ( Homf ` C ) = ( Homf ` O ) )
5 1 3 4 oppcthinco 
 |-  ( ph -> ( comf ` C ) = ( comf ` O ) )