Theorem yonedalem21 16913

Description: Lemma for yoneda 16923. (Contributed by Mario Carneiro, 28-Jan-2017.)

Hypotheses

Ref	Expression
yoneda.y	|- Y = (Yon ` C )
yoneda.b	|- B = ( Base ` C )
yoneda.1	|- .1. = ( Id ` C )
yoneda.o	|- O = (oppCat ` C )
yoneda.s	|- S = ( SetCat ` U )
yoneda.t	|- T = ( SetCat ` V )
yoneda.q	|- Q = ( O FuncCat S )
yoneda.h	|- H = (HomF ` Q )
yoneda.r	|- R = ( ( Q X.c O ) FuncCat T )
yoneda.e	|- E = ( O evalF S )
yoneda.z	|- Z = ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) )
yoneda.c	|- ( ph -> C e. Cat )
yoneda.w	|- ( ph -> V e. W )
yoneda.u	|- ( ph -> ran ( Hom f ` C ) C_ U )
yoneda.v	|- ( ph -> ( ran ( Hom f ` Q ) u. U ) C_ V )
yonedalem21.f	|- ( ph -> F e. ( O Func S ) )
yonedalem21.x	|- ( ph -> X e. B )

Assertion

Ref	Expression
yonedalem21	|- ( ph -> ( F ( 1st ` Z ) X ) = ( ( ( 1st ` Y ) ` X ) ( O Nat S ) F ) )

Proof of Theorem yonedalem21

Step	Hyp	Ref	Expression
1		yoneda.z	. . . . . 6 |- Z = ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) )
2	1	fveq2i 6194	. . . . 5 |- ( 1st ` Z ) = ( 1st ` ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) )
3	2	oveqi 6663	. . . 4 |- ( F ( 1st ` Z ) X ) = ( F ( 1st ` ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ) X )
4		df-ov 6653	. . . 4 |- ( F ( 1st ` ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ) X ) = ( ( 1st ` ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ) ` <. F , X >. )
5	3, 4	eqtri 2644	. . 3 |- ( F ( 1st ` Z ) X ) = ( ( 1st ` ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ) ` <. F , X >. )
6		eqid 2622	. . . . 5 |- ( Q X.c O ) = ( Q X.c O )
7		yoneda.q	. . . . . 6 |- Q = ( O FuncCat S )
8	7	fucbas 16620	. . . . 5 |- ( O Func S ) = ( Base ` Q )
9		yoneda.o	. . . . . 6 |- O = (oppCat ` C )
10		yoneda.b	. . . . . 6 |- B = ( Base ` C )
11	9, 10	oppcbas 16378	. . . . 5 |- B = ( Base ` O )
12	6, 8, 11	xpcbas 16818	. . . 4 |- ( ( O Func S ) X. B ) = ( Base ` ( Q X.c O ) )
13		eqid 2622	. . . . 5 |- ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) = ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) )
14		eqid 2622	. . . . 5 |- ( (oppCat ` Q ) X.c Q ) = ( (oppCat ` Q ) X.c Q )
15		yoneda.c	. . . . . . . . 9 |- ( ph -> C e. Cat )
16	9	oppccat 16382	. . . . . . . . 9 |- ( C e. Cat -> O e. Cat )
17	15, 16	syl 17	. . . . . . . 8 |- ( ph -> O e. Cat )
18		yoneda.w	. . . . . . . . . 10 |- ( ph -> V e. W )
19		yoneda.v	. . . . . . . . . . 11 |- ( ph -> ( ran ( Hom f ` Q ) u. U ) C_ V )
20	19	unssbd 3791	. . . . . . . . . 10 |- ( ph -> U C_ V )
21	18, 20	ssexd 4805	. . . . . . . . 9 |- ( ph -> U e. _V )
22		yoneda.s	. . . . . . . . . 10 |- S = ( SetCat ` U )
23	22	setccat 16735	. . . . . . . . 9 |- ( U e. _V -> S e. Cat )
24	21, 23	syl 17	. . . . . . . 8 |- ( ph -> S e. Cat )
25	7, 17, 24	fuccat 16630	. . . . . . 7 |- ( ph -> Q e. Cat )
26		eqid 2622	. . . . . . 7 |- ( Q 2ndF O ) = ( Q 2ndF O )
27	6, 25, 17, 26	2ndfcl 16838	. . . . . 6 |- ( ph -> ( Q 2ndF O ) e. ( ( Q X.c O ) Func O ) )
28		eqid 2622	. . . . . . . 8 |- (oppCat ` Q ) = (oppCat ` Q )
29		relfunc 16522	. . . . . . . . 9 |- Rel ( C Func Q )
30		yoneda.y	. . . . . . . . . 10 |- Y = (Yon ` C )
31		yoneda.u	. . . . . . . . . 10 |- ( ph -> ran ( Hom f ` C ) C_ U )
32	30, 15, 9, 22, 7, 21, 31	yoncl 16902	. . . . . . . . 9 |- ( ph -> Y e. ( C Func Q ) )
33		1st2ndbr 7217	. . . . . . . . 9 |- ( ( Rel ( C Func Q ) /\ Y e. ( C Func Q ) ) -> ( 1st ` Y ) ( C Func Q ) ( 2nd ` Y ) )
34	29, 32, 33	sylancr 695	. . . . . . . 8 |- ( ph -> ( 1st ` Y ) ( C Func Q ) ( 2nd ` Y ) )
35	9, 28, 34	funcoppc 16535	. . . . . . 7 |- ( ph -> ( 1st ` Y ) ( O Func (oppCat ` Q ) )tpos ( 2nd ` Y ) )
36		df-br 4654	. . . . . . 7 |- ( ( 1st ` Y ) ( O Func (oppCat ` Q ) )tpos ( 2nd ` Y ) <-> <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. e. ( O Func (oppCat ` Q ) ) )
37	35, 36	sylib 208	. . . . . 6 |- ( ph -> <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. e. ( O Func (oppCat ` Q ) ) )
38	27, 37	cofucl 16548	. . . . 5 |- ( ph -> ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) e. ( ( Q X.c O ) Func (oppCat ` Q ) ) )
39		eqid 2622	. . . . . 6 |- ( Q 1stF O ) = ( Q 1stF O )
40	6, 25, 17, 39	1stfcl 16837	. . . . 5 |- ( ph -> ( Q 1stF O ) e. ( ( Q X.c O ) Func Q ) )
41	13, 14, 38, 40	prfcl 16843	. . . 4 |- ( ph -> ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) e. ( ( Q X.c O ) Func ( (oppCat ` Q ) X.c Q ) ) )
42		yoneda.h	. . . . 5 |- H = (HomF ` Q )
43		yoneda.t	. . . . 5 |- T = ( SetCat ` V )
44	19	unssad 3790	. . . . 5 |- ( ph -> ran ( Hom f ` Q ) C_ V )
45	42, 28, 43, 25, 18, 44	hofcl 16899	. . . 4 |- ( ph -> H e. ( ( (oppCat ` Q ) X.c Q ) Func T ) )
46		yonedalem21.f	. . . . 5 |- ( ph -> F e. ( O Func S ) )
47		yonedalem21.x	. . . . 5 |- ( ph -> X e. B )
48		opelxpi 5148	. . . . 5 |- ( ( F e. ( O Func S ) /\ X e. B ) -> <. F , X >. e. ( ( O Func S ) X. B ) )
49	46, 47, 48	syl2anc 693	. . . 4 |- ( ph -> <. F , X >. e. ( ( O Func S ) X. B ) )
50	12, 41, 45, 49	cofu1 16544	. . 3 |- ( ph -> ( ( 1st ` ( H o.func ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ) ` <. F , X >. ) = ( ( 1st ` H ) ` ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. F , X >. ) ) )
51	5, 50	syl5eq 2668	. 2 |- ( ph -> ( F ( 1st ` Z ) X ) = ( ( 1st ` H ) ` ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. F , X >. ) ) )
52		eqid 2622	. . . . . 6 |- ( Hom ` ( Q X.c O ) ) = ( Hom ` ( Q X.c O ) )
53	13, 12, 52, 38, 40, 49	prf1 16840	. . . . 5 |- ( ph -> ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. F , X >. ) = <. ( ( 1st ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) ` <. F , X >. ) , ( ( 1st ` ( Q 1stF O ) ) ` <. F , X >. ) >. )
54	12, 27, 37, 49	cofu1 16544	. . . . . . 7 |- ( ph -> ( ( 1st ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) ` <. F , X >. ) = ( ( 1st ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) ` ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) ) )
55		fvex 6201	. . . . . . . . . 10 |- ( 1st ` Y ) e. _V
56		fvex 6201	. . . . . . . . . . 11 |- ( 2nd ` Y ) e. _V
57	56	tposex 7386	. . . . . . . . . 10 |- tpos ( 2nd ` Y ) e. _V
58	55, 57	op1st 7176	. . . . . . . . 9 |- ( 1st ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) = ( 1st ` Y )
59	58	a1i 11	. . . . . . . 8 |- ( ph -> ( 1st ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) = ( 1st ` Y ) )
60	6, 12, 52, 25, 17, 26, 49	2ndf1 16835	. . . . . . . . 9 |- ( ph -> ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) = ( 2nd ` <. F , X >. ) )
61		op2ndg 7181	. . . . . . . . . 10 |- ( ( F e. ( O Func S ) /\ X e. B ) -> ( 2nd ` <. F , X >. ) = X )
62	46, 47, 61	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( 2nd ` <. F , X >. ) = X )
63	60, 62	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) = X )
64	59, 63	fveq12d 6197	. . . . . . 7 |- ( ph -> ( ( 1st ` <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. ) ` ( ( 1st ` ( Q 2ndF O ) ) ` <. F , X >. ) ) = ( ( 1st ` Y ) ` X ) )
65	54, 64	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( 1st ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) ` <. F , X >. ) = ( ( 1st ` Y ) ` X ) )
66	6, 12, 52, 25, 17, 39, 49	1stf1 16832	. . . . . . 7 |- ( ph -> ( ( 1st ` ( Q 1stF O ) ) ` <. F , X >. ) = ( 1st ` <. F , X >. ) )
67		op1stg 7180	. . . . . . . 8 |- ( ( F e. ( O Func S ) /\ X e. B ) -> ( 1st ` <. F , X >. ) = F )
68	46, 47, 67	syl2anc 693	. . . . . . 7 |- ( ph -> ( 1st ` <. F , X >. ) = F )
69	66, 68	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( 1st ` ( Q 1stF O ) ) ` <. F , X >. ) = F )
70	65, 69	opeq12d 4410	. . . . 5 |- ( ph -> <. ( ( 1st ` ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ) ` <. F , X >. ) , ( ( 1st ` ( Q 1stF O ) ) ` <. F , X >. ) >. = <. ( ( 1st ` Y ) ` X ) , F >. )
71	53, 70	eqtrd 2656	. . . 4 |- ( ph -> ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. F , X >. ) = <. ( ( 1st ` Y ) ` X ) , F >. )
72	71	fveq2d 6195	. . 3 |- ( ph -> ( ( 1st ` H ) ` ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. F , X >. ) ) = ( ( 1st ` H ) ` <. ( ( 1st ` Y ) ` X ) , F >. ) )
73		df-ov 6653	. . 3 |- ( ( ( 1st ` Y ) ` X ) ( 1st ` H ) F ) = ( ( 1st ` H ) ` <. ( ( 1st ` Y ) ` X ) , F >. )
74	72, 73	syl6eqr 2674	. 2 |- ( ph -> ( ( 1st ` H ) ` ( ( 1st ` ( ( <. ( 1st ` Y ) , tpos ( 2nd ` Y ) >. o.func ( Q 2ndF O ) ) ⟨,⟩F ( Q 1stF O ) ) ) ` <. F , X >. ) ) = ( ( ( 1st ` Y ) ` X ) ( 1st ` H ) F ) )
75		eqid 2622	. . . 4 |- ( O Nat S ) = ( O Nat S )
76	7, 75	fuchom 16621	. . 3 |- ( O Nat S ) = ( Hom ` Q )
77	30, 10, 15, 47, 9, 22, 21, 31	yon1cl 16903	. . 3 |- ( ph -> ( ( 1st ` Y ) ` X ) e. ( O Func S ) )
78	42, 25, 8, 76, 77, 46	hof1 16894	. 2 |- ( ph -> ( ( ( 1st ` Y ) ` X ) ( 1st ` H ) F ) = ( ( ( 1st ` Y ) ` X ) ( O Nat S ) F ) )
79	51, 74, 78	3eqtrd 2660	1 |- ( ph -> ( F ( 1st ` Z ) X ) = ( ( ( 1st ` Y ) ` X ) ( O Nat S ) F ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   <.cop 4183   class class class wbr 4653   X. cxp 5112   ran crn 5115   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167  tpos ctpos 7351   Basecbs 15857   Hom chom 15952   Catccat 16325   Idccid 16326   Hom _f chomf 16327  oppCatcoppc 16371   Func cfunc 16514   o._func ccofu 16516   Nat cnat 16601   FuncCat cfuc 16602   SetCatcsetc 16725   X._c cxpc 16808   1st_F c1stf 16809   2nd_F c2ndf 16810   ⟨,⟩_F cprf 16811   eval_F cevlf 16849  Hom_Fchof 16888  Yoncyon 16889

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-homf 16331  df-comf 16332  df-oppc 16372  df-func 16518  df-cofu 16520  df-nat 16603  df-fuc 16604  df-setc 16726  df-xpc 16812  df-1stf 16813  df-2ndf 16814  df-prf 16815  df-curf 16854  df-hof 16890  df-yon 16891

This theorem is referenced by:  yonedalem3a  16914  yonedalem3b  16919  yonedainv  16921  yonffthlem  16922