Theorem cidval 16338

Description: Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
cidfval.b	|- B = ( Base ` C )
cidfval.h	|- H = ( Hom ` C )
cidfval.o	|- .x. = (comp ` C )
cidfval.c	|- ( ph -> C e. Cat )
cidfval.i	|- .1. = ( Id ` C )
cidval.x	|- ( ph -> X e. B )

Assertion

Ref	Expression
cidval	|- ( ph -> ( .1. ` X ) = ( iota_ g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) ) )

Distinct variable groups:   f, g, y, B   C, f, g, y   .x. , f, g, y   f, H, g, y   ph, f, g, y   f, X, g, y

Allowed substitution hints:   .1. ( y, f, g)

Proof of Theorem cidval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cidfval.b	. . 3 |- B = ( Base ` C )
2		cidfval.h	. . 3 |- H = ( Hom ` C )
3		cidfval.o	. . 3 |- .x. = (comp ` C )
4		cidfval.c	. . 3 |- ( ph -> C e. Cat )
5		cidfval.i	. . 3 |- .1. = ( Id ` C )
6	1, 2, 3, 4, 5	cidfval 16337	. 2 |- ( ph -> .1. = ( x e. B |-> ( iota_ g e. ( x H x ) A. y e. B ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f /\ A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f ) ) ) )
7		simpr 477	. . . 4 |- ( ( ph /\ x = X ) -> x = X )
8	7, 7	oveq12d 6668	. . 3 |- ( ( ph /\ x = X ) -> ( x H x ) = ( X H X ) )
9	7	oveq2d 6666	. . . . . 6 |- ( ( ph /\ x = X ) -> ( y H x ) = ( y H X ) )
10	7	opeq2d 4409	. . . . . . . . 9 |- ( ( ph /\ x = X ) -> <. y , x >. = <. y , X >. )
11	10, 7	oveq12d 6668	. . . . . . . 8 |- ( ( ph /\ x = X ) -> ( <. y , x >. .x. x ) = ( <. y , X >. .x. X ) )
12	11	oveqd 6667	. . . . . . 7 |- ( ( ph /\ x = X ) -> ( g ( <. y , x >. .x. x ) f ) = ( g ( <. y , X >. .x. X ) f ) )
13	12	eqeq1d 2624	. . . . . 6 |- ( ( ph /\ x = X ) -> ( ( g ( <. y , x >. .x. x ) f ) = f <-> ( g ( <. y , X >. .x. X ) f ) = f ) )
14	9, 13	raleqbidv 3152	. . . . 5 |- ( ( ph /\ x = X ) -> ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f <-> A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f ) )
15	7	oveq1d 6665	. . . . . 6 |- ( ( ph /\ x = X ) -> ( x H y ) = ( X H y ) )
16	7, 7	opeq12d 4410	. . . . . . . . 9 |- ( ( ph /\ x = X ) -> <. x , x >. = <. X , X >. )
17	16	oveq1d 6665	. . . . . . . 8 |- ( ( ph /\ x = X ) -> ( <. x , x >. .x. y ) = ( <. X , X >. .x. y ) )
18	17	oveqd 6667	. . . . . . 7 |- ( ( ph /\ x = X ) -> ( f ( <. x , x >. .x. y ) g ) = ( f ( <. X , X >. .x. y ) g ) )
19	18	eqeq1d 2624	. . . . . 6 |- ( ( ph /\ x = X ) -> ( ( f ( <. x , x >. .x. y ) g ) = f <-> ( f ( <. X , X >. .x. y ) g ) = f ) )
20	15, 19	raleqbidv 3152	. . . . 5 |- ( ( ph /\ x = X ) -> ( A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f <-> A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) )
21	14, 20	anbi12d 747	. . . 4 |- ( ( ph /\ x = X ) -> ( ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f /\ A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f ) <-> ( A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) ) )
22	21	ralbidv 2986	. . 3 |- ( ( ph /\ x = X ) -> ( A. y e. B ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f /\ A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f ) <-> A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) ) )
23	8, 22	riotaeqbidv 6614	. 2 |- ( ( ph /\ x = X ) -> ( iota_ g e. ( x H x ) A. y e. B ( A. f e. ( y H x ) ( g ( <. y , x >. .x. x ) f ) = f /\ A. f e. ( x H y ) ( f ( <. x , x >. .x. y ) g ) = f ) ) = ( iota_ g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) ) )
24		cidval.x	. 2 |- ( ph -> X e. B )
25		riotaex 6615	. . 3 |- ( iota_ g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) ) e. _V
26	25	a1i 11	. 2 |- ( ph -> ( iota_ g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) ) e. _V )
27	6, 23, 24, 26	fvmptd 6288	1 |- ( ph -> ( .1. ` X ) = ( iota_ g e. ( X H X ) A. y e. B ( A. f e. ( y H X ) ( g ( <. y , X >. .x. X ) f ) = f /\ A. f e. ( X H y ) ( f ( <. X , X >. .x. y ) g ) = f ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326

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-rep 4771  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-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-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-riota 6611  df-ov 6653  df-cid 16330

This theorem is referenced by:  catidcl  16343  catlid  16344  catrid  16345