Theorem idfuval 16536

Description: Value 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 )
idfuval.h	|- H = ( Hom ` C )

Assertion

Ref	Expression
idfuval	|- ( ph -> I = <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) ) >. )

Distinct variable groups:   z, B   z, C   z, H   ph, z

Allowed substitution hint:   I( z)

Proof of Theorem idfuval

Dummy variables b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idfuval.i	. 2 |- I = (idfunc ` C )
2		idfuval.c	. . 3 |- ( ph -> C e. Cat )
3		fvexd 6203	. . . . 5 |- ( c = C -> ( Base ` c ) e. _V )
4		fveq2 6191	. . . . . 6 |- ( c = C -> ( Base ` c ) = ( Base ` C ) )
5		idfuval.b	. . . . . 6 |- B = ( Base ` C )
6	4, 5	syl6eqr 2674	. . . . 5 |- ( c = C -> ( Base ` c ) = B )
7		simpr 477	. . . . . . 7 |- ( ( c = C /\ b = B ) -> b = B )
8	7	reseq2d 5396	. . . . . 6 |- ( ( c = C /\ b = B ) -> ( _I |` b ) = ( _I |` B ) )
9	7	sqxpeqd 5141	. . . . . . 7 |- ( ( c = C /\ b = B ) -> ( b X. b ) = ( B X. B ) )
10		simpl 473	. . . . . . . . . . 11 |- ( ( c = C /\ b = B ) -> c = C )
11	10	fveq2d 6195	. . . . . . . . . 10 |- ( ( c = C /\ b = B ) -> ( Hom ` c ) = ( Hom ` C ) )
12		idfuval.h	. . . . . . . . . 10 |- H = ( Hom ` C )
13	11, 12	syl6eqr 2674	. . . . . . . . 9 |- ( ( c = C /\ b = B ) -> ( Hom ` c ) = H )
14	13	fveq1d 6193	. . . . . . . 8 |- ( ( c = C /\ b = B ) -> ( ( Hom ` c ) ` z ) = ( H ` z ) )
15	14	reseq2d 5396	. . . . . . 7 |- ( ( c = C /\ b = B ) -> ( _I |` ( ( Hom ` c ) ` z ) ) = ( _I |` ( H ` z ) ) )
16	9, 15	mpteq12dv 4733	. . . . . 6 |- ( ( c = C /\ b = B ) -> ( z e. ( b X. b ) |-> ( _I |` ( ( Hom ` c ) ` z ) ) ) = ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) ) )
17	8, 16	opeq12d 4410	. . . . 5 |- ( ( c = C /\ b = B ) -> <. ( _I |` b ) , ( z e. ( b X. b ) |-> ( _I |` ( ( Hom ` c ) ` z ) ) ) >. = <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) ) >. )
18	3, 6, 17	csbied2 3561	. . . 4 |- ( c = C -> [_ ( Base ` c ) / b ]_ <. ( _I |` b ) , ( z e. ( b X. b ) |-> ( _I |` ( ( Hom ` c ) ` z ) ) ) >. = <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) ) >. )
19		df-idfu 16519	. . . 4 |- idfunc = ( c e. Cat |-> [_ ( Base ` c ) / b ]_ <. ( _I |` b ) , ( z e. ( b X. b ) |-> ( _I |` ( ( Hom ` c ) ` z ) ) ) >. )
20		opex 4932	. . . 4 |- <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) ) >. e. _V
21	18, 19, 20	fvmpt 6282	. . 3 |- ( C e. Cat -> (idfunc ` C ) = <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) ) >. )
22	2, 21	syl 17	. 2 |- ( ph -> (idfunc ` C ) = <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) ) >. )
23	1, 22	syl5eq 2668	1 |- ( ph -> I = <. ( _I |` B ) , ( z e. ( B X. B ) |-> ( _I |` ( H ` z ) ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   <.cop 4183   |-> cmpt 4729   _I cid 5023   X. cxp 5112   |` cres 5116   ` cfv 5888   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-idfu 16519

This theorem is referenced by:  idfu2nd  16537  idfu1st  16539  idfucl  16541  catcisolem  16756  curf2ndf  16887  idfusubc0  41865