Theorem yonedalem21 18234

Description: Lemma for yoneda 18244. (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 ×c 𝑂) FuncCat 𝑇)
yoneda.e	⊢ 𝐸 = (𝑂 evalF 𝑆)
yoneda.z	⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c	⊢ (𝜑 → 𝐶 ∈ Cat)
yoneda.w	⊢ (𝜑 → 𝑉 ∈ 𝑊)
yoneda.u	⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
yoneda.v	⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
yonedalem21.f	⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))
yonedalem21.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
yonedalem21	⊢ (𝜑 → (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 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
2	1	fveq2i 6839	. . . . 5 ⊢ (1st ‘𝑍) = (1st ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))
3	2	oveqi 7375	. . . 4 ⊢ (𝐹(1st ‘𝑍)𝑋) = (𝐹(1st ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))𝑋)
4		df-ov 7365	. . . 4 ⊢ (𝐹(1st ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))𝑋) = ((1st ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))‘⟨𝐹, 𝑋⟩)
5	3, 4	eqtri 2760	. . 3 ⊢ (𝐹(1st ‘𝑍)𝑋) = ((1st ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))‘⟨𝐹, 𝑋⟩)
6		eqid 2737	. . . . 5 ⊢ (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
7		yoneda.q	. . . . . 6 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
8	7	fucbas 17925	. . . . 5 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
9		yoneda.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
10		yoneda.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10	oppcbas 17679	. . . . 5 ⊢ 𝐵 = (Base‘𝑂)
12	6, 8, 11	xpcbas 18139	. . . 4 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
13		eqid 2737	. . . . 5 ⊢ ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)) = ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))
14		eqid 2737	. . . . 5 ⊢ ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
15		yoneda.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
16	9	oppccat 17683	. . . . . . . . 9 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
17	15, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ Cat)
18		yoneda.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
19		yoneda.v	. . . . . . . . . . 11 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
20	19	unssbd 4135	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
21	18, 20	ssexd 5262	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
22		yoneda.s	. . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈)
23	22	setccat 18047	. . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
24	21, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat)
25	7, 17, 24	fuccat 17935	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat)
26		eqid 2737	. . . . . . 7 ⊢ (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
27	6, 25, 17, 26	2ndfcl 18159	. . . . . 6 ⊢ (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
28		eqid 2737	. . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄)
29		relfunc 17824	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝑄)
30		yoneda.y	. . . . . . . . . 10 ⊢ 𝑌 = (Yon‘𝐶)
31		yoneda.u	. . . . . . . . . 10 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
32	30, 15, 9, 22, 7, 21, 31	yoncl 18223	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
33		1st2ndbr 7990	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
34	29, 32, 33	sylancr 588	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
35	9, 28, 34	funcoppc 17837	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌))
36		df-br 5087	. . . . . . 7 ⊢ ((1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌) ↔ ⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
37	35, 36	sylib 218	. . . . . 6 ⊢ (𝜑 → ⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
38	27, 37	cofucl 17850	. . . . 5 ⊢ (𝜑 → (⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
39		eqid 2737	. . . . . 6 ⊢ (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
40	6, 25, 17, 39	1stfcl 18158	. . . . 5 ⊢ (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
41	13, 14, 38, 40	prfcl 18164	. . . 4 ⊢ (𝜑 → ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func ((oppCat‘𝑄) ×c 𝑄)))
42		yoneda.h	. . . . 5 ⊢ 𝐻 = (HomF‘𝑄)
43		yoneda.t	. . . . 5 ⊢ 𝑇 = (SetCat‘𝑉)
44	19	unssad 4134	. . . . 5 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉)
45	42, 28, 43, 25, 18, 44	hofcl 18220	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
46		yonedalem21.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))
47		yonedalem21.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
48	46, 47	opelxpd 5665	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
49	12, 41, 45, 48	cofu1 17846	. . 3 ⊢ (𝜑 → ((1st ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))‘⟨𝐹, 𝑋⟩) = ((1st ‘𝐻)‘((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)))
50	5, 49	eqtrid 2784	. 2 ⊢ (𝜑 → (𝐹(1st ‘𝑍)𝑋) = ((1st ‘𝐻)‘((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)))
51		eqid 2737	. . . . . 6 ⊢ (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
52	13, 12, 51, 38, 40, 48	prf1 18161	. . . . 5 ⊢ (𝜑 → ((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩) = ⟨((1st ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩)⟩)
53	12, 27, 37, 48	cofu1 17846	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)))
54		fvex 6849	. . . . . . . . . 10 ⊢ (1st ‘𝑌) ∈ V
55		fvex 6849	. . . . . . . . . . 11 ⊢ (2nd ‘𝑌) ∈ V
56	55	tposex 8205	. . . . . . . . . 10 ⊢ tpos (2nd ‘𝑌) ∈ V
57	54, 56	op1st 7945	. . . . . . . . 9 ⊢ (1st ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩) = (1st ‘𝑌)
58	57	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩) = (1st ‘𝑌))
59	6, 12, 51, 25, 17, 26, 48	2ndf1 18156	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = (2nd ‘⟨𝐹, 𝑋⟩))
60		op2ndg 7950	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
61	46, 47, 60	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
62	59, 61	eqtrd 2772	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
63	58, 62	fveq12d 6843	. . . . . . 7 ⊢ (𝜑 → ((1st ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = ((1st ‘𝑌)‘𝑋))
64	53, 63	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → ((1st ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st ‘𝑌)‘𝑋))
65	6, 12, 51, 25, 17, 39, 48	1stf1 18153	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = (1st ‘⟨𝐹, 𝑋⟩))
66		op1stg 7949	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
67	46, 47, 66	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
68	65, 67	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
69	64, 68	opeq12d 4825	. . . . 5 ⊢ (𝜑 → ⟨((1st ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩)⟩ = ⟨((1st ‘𝑌)‘𝑋), 𝐹⟩)
70	52, 69	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩) = ⟨((1st ‘𝑌)‘𝑋), 𝐹⟩)
71	70	fveq2d 6840	. . 3 ⊢ (𝜑 → ((1st ‘𝐻)‘((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)) = ((1st ‘𝐻)‘⟨((1st ‘𝑌)‘𝑋), 𝐹⟩))
72		df-ov 7365	. . 3 ⊢ (((1st ‘𝑌)‘𝑋)(1st ‘𝐻)𝐹) = ((1st ‘𝐻)‘⟨((1st ‘𝑌)‘𝑋), 𝐹⟩)
73	71, 72	eqtr4di 2790	. 2 ⊢ (𝜑 → ((1st ‘𝐻)‘((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)) = (((1st ‘𝑌)‘𝑋)(1st ‘𝐻)𝐹))
74		eqid 2737	. . . 4 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
75	7, 74	fuchom 17926	. . 3 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
76	30, 10, 15, 47, 9, 22, 21, 31	yon1cl 18224	. . 3 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
77	42, 25, 8, 75, 76, 46	hof1 18215	. 2 ⊢ (𝜑 → (((1st ‘𝑌)‘𝑋)(1st ‘𝐻)𝐹) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
78	50, 73, 77	3eqtrd 2776	1 ⊢ (𝜑 → (𝐹(1st ‘𝑍)𝑋) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∪ cun 3888   ⊆ wss 3890  ⟨cop 4574   class class class wbr 5086   × cxp 5624  ran crn 5627  Rel wrel 5631  ‘cfv 6494  (class class class)co 7362  1^st c1st 7935  2^nd c2nd 7936  tpos ctpos 8170  Basecbs 17174  Hom chom 17226  Catccat 17625  Idccid 17626  Hom_f chomf 17627  oppCatcoppc 17672   Func cfunc 17816   ∘_func ccofu 17818   Nat cnat 17906   FuncCat cfuc 17907  SetCatcsetc 18037   ×_c cxpc 18129   1^st_F c1stf 18130   2^nd_F c2ndf 18131   ⟨,⟩_F cprf 18132   eval_F cevlf 18170  Hom_Fchof 18209  Yoncyon 18210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-tpos 8171  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-map 8770  df-ixp 8841  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-hom 17239  df-cco 17240  df-cat 17629  df-cid 17630  df-homf 17631  df-comf 17632  df-oppc 17673  df-func 17820  df-cofu 17822  df-nat 17908  df-fuc 17909  df-setc 18038  df-xpc 18133  df-1stf 18134  df-2ndf 18135  df-prf 18136  df-curf 18175  df-hof 18211  df-yon 18212

This theorem is referenced by:  yonedalem3a  18235  yonedalem3b  18240  yonedainv  18242  yonffthlem  18243