Theorem idfu1stf1o

Description: The identity functor/inclusion functor is bijective on objects. (Contributed by Zhi Wang, 16-Nov-2025)

	Ref	Expression
Hypotheses	idfu1stf1o.i	|- I = ( idFunc ` C )
	idfu1stf1o.b	|- B = ( Base ` C )
Assertion	idfu1stf1o	|- ( C e. Cat -> ( 1st ` I ) : B -1-1-onto-> B )

Proof

Step	Hyp	Ref	Expression
1		idfu1stf1o.i	|- I = ( idFunc ` C )
2		idfu1stf1o.b	|- B = ( Base ` C )
3		f1oi	|- ( _I |` B ) : B -1-1-onto-> B
4		id	|- ( C e. Cat -> C e. Cat )
5	1 2 4	idfu1st	|- ( C e. Cat -> ( 1st ` I ) = ( _I |` B ) )
6	5	f1oeq1d	|- ( C e. Cat -> ( ( 1st ` I ) : B -1-1-onto-> B <-> ( _I |` B ) : B -1-1-onto-> B ) )
7	3 6	mpbiri	|- ( C e. Cat -> ( 1st ` I ) : B -1-1-onto-> B )