Theorem fucidcl 16625

Description: The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
fucidcl.q	|- Q = ( C FuncCat D )
fucidcl.n	|- N = ( C Nat D )
fucidcl.x	|- .1. = ( Id ` D )
fucidcl.f	|- ( ph -> F e. ( C Func D ) )

Assertion

Ref	Expression
fucidcl	|- ( ph -> ( .1. o. ( 1st ` F ) ) e. ( F N F ) )

Proof of Theorem fucidcl

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

Step	Hyp	Ref	Expression
1		fucidcl.f	. . . . . . . 8 |- ( ph -> F e. ( C Func D ) )
2		funcrcl 16523	. . . . . . . 8 |- ( F e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
3	1, 2	syl 17	. . . . . . 7 |- ( ph -> ( C e. Cat /\ D e. Cat ) )
4	3	simprd 479	. . . . . 6 |- ( ph -> D e. Cat )
5		eqid 2622	. . . . . . 7 |- ( Base ` D ) = ( Base ` D )
6		fucidcl.x	. . . . . . 7 |- .1. = ( Id ` D )
7	5, 6	cidfn 16340	. . . . . 6 |- ( D e. Cat -> .1. Fn ( Base ` D ) )
8	4, 7	syl 17	. . . . 5 |- ( ph -> .1. Fn ( Base ` D ) )
9		dffn2 6047	. . . . 5 |- ( .1. Fn ( Base ` D ) <-> .1. : ( Base ` D ) --> _V )
10	8, 9	sylib 208	. . . 4 |- ( ph -> .1. : ( Base ` D ) --> _V )
11		eqid 2622	. . . . 5 |- ( Base ` C ) = ( Base ` C )
12		relfunc 16522	. . . . . 6 |- Rel ( C Func D )
13		1st2ndbr 7217	. . . . . 6 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
14	12, 1, 13	sylancr 695	. . . . 5 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
15	11, 5, 14	funcf1 16526	. . . 4 |- ( ph -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) )
16		fcompt 6400	. . . 4 |- ( ( .1. : ( Base ` D ) --> _V /\ ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) ) -> ( .1. o. ( 1st ` F ) ) = ( x e. ( Base ` C ) |-> ( .1. ` ( ( 1st ` F ) ` x ) ) ) )
17	10, 15, 16	syl2anc 693	. . 3 |- ( ph -> ( .1. o. ( 1st ` F ) ) = ( x e. ( Base ` C ) |-> ( .1. ` ( ( 1st ` F ) ` x ) ) ) )
18		eqid 2622	. . . . . 6 |- ( Hom ` D ) = ( Hom ` D )
19	4	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> D e. Cat )
20	15	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
21	5, 18, 6, 19, 20	catidcl 16343	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( .1. ` ( ( 1st ` F ) ` x ) ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` x ) ) )
22	21	ralrimiva 2966	. . . 4 |- ( ph -> A. x e. ( Base ` C ) ( .1. ` ( ( 1st ` F ) ` x ) ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` x ) ) )
23		fvex 6201	. . . . 5 |- ( Base ` C ) e. _V
24		mptelixpg 7945	. . . . 5 |- ( ( Base ` C ) e. _V -> ( ( x e. ( Base ` C ) |-> ( .1. ` ( ( 1st ` F ) ` x ) ) ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` x ) ) <-> A. x e. ( Base ` C ) ( .1. ` ( ( 1st ` F ) ` x ) ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` x ) ) ) )
25	23, 24	ax-mp 5	. . . 4 |- ( ( x e. ( Base ` C ) |-> ( .1. ` ( ( 1st ` F ) ` x ) ) ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` x ) ) <-> A. x e. ( Base ` C ) ( .1. ` ( ( 1st ` F ) ` x ) ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` x ) ) )
26	22, 25	sylibr 224	. . 3 |- ( ph -> ( x e. ( Base ` C ) |-> ( .1. ` ( ( 1st ` F ) ` x ) ) ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` x ) ) )
27	17, 26	eqeltrd 2701	. 2 |- ( ph -> ( .1. o. ( 1st ` F ) ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` x ) ) )
28	4	adantr 481	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> D e. Cat )
29		simpr1 1067	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> x e. ( Base ` C ) )
30	29, 20	syldan 487	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
31		eqid 2622	. . . . . 6 |- (comp ` D ) = (comp ` D )
32	15	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) )
33		simpr2 1068	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> y e. ( Base ` C ) )
34	32, 33	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` F ) ` y ) e. ( Base ` D ) )
35		eqid 2622	. . . . . . . 8 |- ( Hom ` C ) = ( Hom ` C )
36	14	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
37	11, 35, 18, 36, 29, 33	funcf2 16528	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) )
38		simpr3 1069	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> f e. ( x ( Hom ` C ) y ) )
39	37, 38	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( x ( 2nd ` F ) y ) ` f ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) )
40	5, 18, 6, 28, 30, 31, 34, 39	catlid 16344	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( .1. ` ( ( 1st ` F ) ` y ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( x ( 2nd ` F ) y ) ` f ) )
41	5, 18, 6, 28, 30, 31, 34, 39	catrid 16345	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( x ( 2nd ` F ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` x ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( .1. ` ( ( 1st ` F ) ` x ) ) ) = ( ( x ( 2nd ` F ) y ) ` f ) )
42	40, 41	eqtr4d 2659	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( .1. ` ( ( 1st ` F ) ` y ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` F ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` x ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( .1. ` ( ( 1st ` F ) ` x ) ) ) )
43		fvco3 6275	. . . . . 6 |- ( ( ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) /\ y e. ( Base ` C ) ) -> ( ( .1. o. ( 1st ` F ) ) ` y ) = ( .1. ` ( ( 1st ` F ) ` y ) ) )
44	32, 33, 43	syl2anc 693	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( .1. o. ( 1st ` F ) ) ` y ) = ( .1. ` ( ( 1st ` F ) ` y ) ) )
45	44	oveq1d 6665	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( .1. o. ( 1st ` F ) ) ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( .1. ` ( ( 1st ` F ) ` y ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) )
46		fvco3 6275	. . . . . 6 |- ( ( ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) /\ x e. ( Base ` C ) ) -> ( ( .1. o. ( 1st ` F ) ) ` x ) = ( .1. ` ( ( 1st ` F ) ` x ) ) )
47	32, 29, 46	syl2anc 693	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( .1. o. ( 1st ` F ) ) ` x ) = ( .1. ` ( ( 1st ` F ) ` x ) ) )
48	47	oveq2d 6666	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( x ( 2nd ` F ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` x ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( ( .1. o. ( 1st ` F ) ) ` x ) ) = ( ( ( x ( 2nd ` F ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` x ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( .1. ` ( ( 1st ` F ) ` x ) ) ) )
49	42, 45, 48	3eqtr4d 2666	. . 3 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( .1. o. ( 1st ` F ) ) ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` F ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` x ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( ( .1. o. ( 1st ` F ) ) ` x ) ) )
50	49	ralrimivvva 2972	. 2 |- ( ph -> A. x e. ( Base ` C ) A. y e. ( Base ` C ) A. f e. ( x ( Hom ` C ) y ) ( ( ( .1. o. ( 1st ` F ) ) ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` F ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` x ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( ( .1. o. ( 1st ` F ) ) ` x ) ) )
51		fucidcl.n	. . 3 |- N = ( C Nat D )
52	51, 11, 35, 18, 31, 1, 1	isnat2 16608	. 2 |- ( ph -> ( ( .1. o. ( 1st ` F ) ) e. ( F N F ) <-> ( ( .1. o. ( 1st ` F ) ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` x ) ) /\ A. x e. ( Base ` C ) A. y e. ( Base ` C ) A. f e. ( x ( Hom ` C ) y ) ( ( ( .1. o. ( 1st ` F ) ) ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` F ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` x ) >. (comp ` D ) ( ( 1st ` F ) ` y ) ) ( ( .1. o. ( 1st ` F ) ) ` x ) ) ) ) )
53	27, 50, 52	mpbir2and 957	1 |- ( ph -> ( .1. o. ( 1st ` F ) ) e. ( F N F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   o. ccom 5118   Rel wrel 5119   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   X_cixp 7908   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326   Func cfunc 16514   Nat cnat 16601   FuncCat cfuc 16602

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-nat 16603

This theorem is referenced by:  fuclid  16626  fucrid  16627  fuccatid  16629