Theorem idfu1st 16539

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

Hypotheses

Ref	Expression
idfuval.i	|- I = (idfunc ` C )
idfuval.b	|- B = ( Base ` C )
idfuval.c	|- ( ph -> C e. Cat )

Assertion

Ref	Expression
idfu1st	|- ( ph -> ( 1st ` I ) = ( _I |` B ) )

Proof of Theorem idfu1st

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idfuval.i	. . . 4 |- I = (idfunc ` C )
2		idfuval.b	. . . 4 |- B = ( Base ` C )
3		idfuval.c	. . . 4 |- ( ph -> C e. Cat )
4		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
5	1, 2, 3, 4	idfuval 16536	. . 3 |- ( ph -> I = <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( ( Hom ` C ) ` z ) ) ) >. )
6	5	fveq2d 6195	. 2 |- ( ph -> ( 1st ` I ) = ( 1st ` <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( ( Hom ` C ) ` z ) ) ) >. ) )
7		fvex 6201	. . . . 5 |- ( Base ` C ) e. _V
8	2, 7	eqeltri 2697	. . . 4 |- B e. _V
9		resiexg 7102	. . . 4 |- ( B e. _V -> ( _I |` B ) e. _V )
10	8, 9	ax-mp 5	. . 3 |- ( _I |` B ) e. _V
11	8, 8	xpex 6962	. . . 4 |- ( B X. B ) e. _V
12	11	mptex 6486	. . 3 |- ( z e. ( B X. B ) |-> ( _I |` ( ( Hom ` C ) ` z ) ) ) e. _V
13	10, 12	op1st 7176	. 2 |- ( 1st ` <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( ( Hom ` C ) ` z ) ) ) >. ) = ( _I |` B )
14	6, 13	syl6eq 2672	1 |- ( ph -> ( 1st ` I ) = ( _I |` B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   _I cid 5023   X. cxp 5112   |` cres 5116   ` cfv 5888   1stc1st 7166   Basecbs 15857   Hom chom 15952   Catccat 16325  id_funccidfu 16515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-idfu 16519

This theorem is referenced by:  idfu1  16540  cofulid  16550  cofurid  16551  catciso  16757  curf2ndf  16887