Theorem iszeroo

Description: The predicate "is a zero object" of a category. (Contributed by AV, 3-Apr-2020)

	Ref	Expression
Hypotheses	isinito.b	|- B = ( Base ` C )
	isinito.h	|- H = ( Hom ` C )
	isinito.c	|- ( ph -> C e. Cat )
	isinito.i	|- ( ph -> I e. B )
Assertion	iszeroo	|- ( ph -> ( I e. ( ZeroO ` C ) <-> ( I e. ( InitO ` C ) /\ I e. ( TermO ` C ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		isinito.b	|- B = ( Base ` C )
2		isinito.h	|- H = ( Hom ` C )
3		isinito.c	|- ( ph -> C e. Cat )
4		isinito.i	|- ( ph -> I e. B )
5	3 1 2	zerooval	|- ( ph -> ( ZeroO ` C ) = ( ( InitO ` C ) i^i ( TermO ` C ) ) )
6	5	eleq2d	|- ( ph -> ( I e. ( ZeroO ` C ) <-> I e. ( ( InitO ` C ) i^i ( TermO ` C ) ) ) )
7		elin	|- ( I e. ( ( InitO ` C ) i^i ( TermO ` C ) ) <-> ( I e. ( InitO ` C ) /\ I e. ( TermO ` C ) ) )
8	6 7	bitrdi	|- ( ph -> ( I e. ( ZeroO ` C ) <-> ( I e. ( InitO ` C ) /\ I e. ( TermO ` C ) ) ) )