Theorem cofulid 16550

Description: The identity functor is a left identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
cofulid.g	|- ( ph -> F e. ( C Func D ) )
cofulid.1	|- I = (idfunc ` D )

Assertion

Ref	Expression
cofulid	|- ( ph -> ( I o.func F ) = F )

Proof of Theorem cofulid

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cofulid.1	. . . . . 6 |- I = (idfunc ` D )
2		eqid 2622	. . . . . 6 |- ( Base ` D ) = ( Base ` D )
3		cofulid.g	. . . . . . . 8 |- ( ph -> F e. ( C Func D ) )
4		funcrcl 16523	. . . . . . . 8 |- ( F e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
5	3, 4	syl 17	. . . . . . 7 |- ( ph -> ( C e. Cat /\ D e. Cat ) )
6	5	simprd 479	. . . . . 6 |- ( ph -> D e. Cat )
7	1, 2, 6	idfu1st 16539	. . . . 5 |- ( ph -> ( 1st ` I ) = ( _I |` ( Base ` D ) ) )
8	7	coeq1d 5283	. . . 4 |- ( ph -> ( ( 1st ` I ) o. ( 1st ` F ) ) = ( ( _I |` ( Base ` D ) ) o. ( 1st ` F ) ) )
9		eqid 2622	. . . . . 6 |- ( Base ` C ) = ( Base ` C )
10		relfunc 16522	. . . . . . 7 |- Rel ( C Func D )
11		1st2ndbr 7217	. . . . . . 7 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
12	10, 3, 11	sylancr 695	. . . . . 6 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
13	9, 2, 12	funcf1 16526	. . . . 5 |- ( ph -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) )
14		fcoi2 6079	. . . . 5 |- ( ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) -> ( ( _I |` ( Base ` D ) ) o. ( 1st ` F ) ) = ( 1st ` F ) )
15	13, 14	syl 17	. . . 4 |- ( ph -> ( ( _I |` ( Base ` D ) ) o. ( 1st ` F ) ) = ( 1st ` F ) )
16	8, 15	eqtrd 2656	. . 3 |- ( ph -> ( ( 1st ` I ) o. ( 1st ` F ) ) = ( 1st ` F ) )
17	6	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> D e. Cat )
18		eqid 2622	. . . . . . . 8 |- ( Hom ` D ) = ( Hom ` D )
19	13	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
20	19	3adant3 1081	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
21	13	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ y e. ( Base ` C ) ) -> ( ( 1st ` F ) ` y ) e. ( Base ` D ) )
22	21	3adant2 1080	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( 1st ` F ) ` y ) e. ( Base ` D ) )
23	1, 2, 17, 18, 20, 22	idfu2nd 16537	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( ( 1st ` F ) ` x ) ( 2nd ` I ) ( ( 1st ` F ) ` y ) ) = ( _I |` ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) ) )
24	23	coeq1d 5283	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` I ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) = ( ( _I |` ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) ) o. ( x ( 2nd ` F ) y ) ) )
25		eqid 2622	. . . . . . . 8 |- ( Hom ` C ) = ( Hom ` C )
26	12	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
27		simp2 1062	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> x e. ( Base ` C ) )
28		simp3 1063	. . . . . . . 8 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> y e. ( Base ` C ) )
29	9, 25, 18, 26, 27, 28	funcf2 16528	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) )
30		fcoi2 6079	. . . . . . 7 |- ( ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) -> ( ( _I |` ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) ) o. ( x ( 2nd ` F ) y ) ) = ( x ( 2nd ` F ) y ) )
31	29, 30	syl 17	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( _I |` ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) ) o. ( x ( 2nd ` F ) y ) ) = ( x ( 2nd ` F ) y ) )
32	24, 31	eqtrd 2656	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) /\ y e. ( Base ` C ) ) -> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` I ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) = ( x ( 2nd ` F ) y ) )
33	32	mpt2eq3dva 6719	. . . 4 |- ( ph -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` I ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` F ) y ) ) )
34	9, 12	funcfn2 16529	. . . . 5 |- ( ph -> ( 2nd ` F ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
35		fnov 6768	. . . . 5 |- ( ( 2nd ` F ) Fn ( ( Base ` C ) X. ( Base ` C ) ) <-> ( 2nd ` F ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` F ) y ) ) )
36	34, 35	sylib 208	. . . 4 |- ( ph -> ( 2nd ` F ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x ( 2nd ` F ) y ) ) )
37	33, 36	eqtr4d 2659	. . 3 |- ( ph -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` I ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) ) = ( 2nd ` F ) )
38	16, 37	opeq12d 4410	. 2 |- ( ph -> <. ( ( 1st ` I ) o. ( 1st ` F ) ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` I ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) ) >. = <. ( 1st ` F ) , ( 2nd ` F ) >. )
39	1	idfucl 16541	. . . 4 |- ( D e. Cat -> I e. ( D Func D ) )
40	6, 39	syl 17	. . 3 |- ( ph -> I e. ( D Func D ) )
41	9, 3, 40	cofuval 16542	. 2 |- ( ph -> ( I o.func F ) = <. ( ( 1st ` I ) o. ( 1st ` F ) ) , ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` I ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) ) >. )
42		1st2nd 7214	. . 3 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
43	10, 3, 42	sylancr 695	. 2 |- ( ph -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
44	38, 41, 43	3eqtr4d 2666	1 |- ( ph -> ( I o.func F ) = F )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   _I cid 5023   X. cxp 5112   |` cres 5116   o. ccom 5118   Rel wrel 5119   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952   Catccat 16325   Func cfunc 16514  id_funccidfu 16515   o._func ccofu 16516

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-rmo 2920  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-cat 16329  df-cid 16330  df-func 16518  df-idfu 16519  df-cofu 16520

This theorem is referenced by:  catccatid  16752