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Metamath Proof Explorer

Theorem zerooval

Description: The value of the zero object function, i.e. the set of all zero objects of a category. (Contributed by AV, 3-Apr-2020)

Ref Expression
Hypotheses initoval.c 
|- ( ph -> C e. Cat )
initoval.b 
|- B = ( Base ` C )
initoval.h 
|- H = ( Hom ` C )
Assertion zerooval 
|- ( ph -> ( ZeroO ` C ) = ( ( InitO ` C ) i^i ( TermO ` C ) ) )

Proof

Step Hyp Ref Expression
1 initoval.c 
 |-  ( ph -> C e. Cat )
2 initoval.b 
 |-  B = ( Base ` C )
3 initoval.h 
 |-  H = ( Hom ` C )
4 df-zeroo 
 |-  ZeroO = ( c e. Cat |-> ( ( InitO ` c ) i^i ( TermO ` c ) ) )
5 fveq2 
 |-  ( c = C -> ( InitO ` c ) = ( InitO ` C ) )
6 fveq2 
 |-  ( c = C -> ( TermO ` c ) = ( TermO ` C ) )
7 5 6 ineq12d 
 |-  ( c = C -> ( ( InitO ` c ) i^i ( TermO ` c ) ) = ( ( InitO ` C ) i^i ( TermO ` C ) ) )
8 fvex 
 |-  ( InitO ` C ) e. _V
9 8 inex1 
 |-  ( ( InitO ` C ) i^i ( TermO ` C ) ) e. _V
10 9 a1i 
 |-  ( ph -> ( ( InitO ` C ) i^i ( TermO ` C ) ) e. _V )
11 4 7 1 10 fvmptd3 
 |-  ( ph -> ( ZeroO ` C ) = ( ( InitO ` C ) i^i ( TermO ` C ) ) )