Theorem idfu1st

Description: Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017)

	Ref	Expression
Hypotheses	idfuval.i	|- I = ( idFunc ` C )
	idfuval.b	|- B = ( Base ` C )
	idfuval.c	|- ( ph -> C e. Cat )
Assertion	idfu1st	|- ( ph -> ( 1st ` I ) = ( _I |` B ) )

Proof

Step	Hyp	Ref	Expression
1		idfuval.i	|- I = ( idFunc ` C )
2		idfuval.b	|- B = ( Base ` C )
3		idfuval.c	|- ( ph -> C e. Cat )
4		eqid	|- ( Hom ` C ) = ( Hom ` C )
5	1 2 3 4	idfuval	|- ( ph -> I = <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( ( Hom ` C ) ` z ) ) ) >. )
6	5	fveq2d	|- ( ph -> ( 1st ` I ) = ( 1st ` <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( ( Hom ` C ) ` z ) ) ) >. ) )
7	2	fvexi	|- B e. _V
8		resiexg	|- ( B e. _V -> ( _I |` B ) e. _V )
9	7 8	ax-mp	|- ( _I |` B ) e. _V
10	7 7	xpex	|- ( B X. B ) e. _V
11	10	mptex	|- ( z e. ( B X. B ) |-> ( _I |` ( ( Hom ` C ) ` z ) ) ) e. _V
12	9 11	op1st	|- ( 1st ` <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( ( Hom ` C ) ` z ) ) ) >. ) = ( _I |` B )
13	6 12	eqtrdi	|- ( ph -> ( 1st ` I ) = ( _I |` B ) )