Theorem yonedalem21 18285

Description: Lemma for yoneda 18295. (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 6879	. . . . 5 ⊢ (1st ‘𝑍) = (1st ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))
3	2	oveqi 7418	. . . 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 7408	. . . 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 2758	. . 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 2735	. . . . 5 ⊢ (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
7		yoneda.q	. . . . . 6 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
8	7	fucbas 17976	. . . . 5 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
9		yoneda.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
10		yoneda.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10	oppcbas 17730	. . . . 5 ⊢ 𝐵 = (Base‘𝑂)
12	6, 8, 11	xpcbas 18190	. . . 4 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
13		eqid 2735	. . . . 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 2735	. . . . 5 ⊢ ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
15		yoneda.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
16	9	oppccat 17734	. . . . . . . . 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 4169	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
21	18, 20	ssexd 5294	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
22		yoneda.s	. . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈)
23	22	setccat 18098	. . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
24	21, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat)
25	7, 17, 24	fuccat 17986	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat)
26		eqid 2735	. . . . . . 7 ⊢ (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
27	6, 25, 17, 26	2ndfcl 18210	. . . . . 6 ⊢ (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
28		eqid 2735	. . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄)
29		relfunc 17875	. . . . . . . . 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 18274	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
33		1st2ndbr 8041	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
34	29, 32, 33	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
35	9, 28, 34	funcoppc 17888	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌))
36		df-br 5120	. . . . . . 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 17901	. . . . 5 ⊢ (𝜑 → (⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
39		eqid 2735	. . . . . 6 ⊢ (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
40	6, 25, 17, 39	1stfcl 18209	. . . . 5 ⊢ (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
41	13, 14, 38, 40	prfcl 18215	. . . 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 4168	. . . . 5 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉)
45	42, 28, 43, 25, 18, 44	hofcl 18271	. . . 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 5693	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
49	12, 41, 45, 48	cofu1 17897	. . 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 2782	. 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 2735	. . . . . 6 ⊢ (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
52	13, 12, 51, 38, 40, 48	prf1 18212	. . . . 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 17897	. . . . . . 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 6889	. . . . . . . . . 10 ⊢ (1st ‘𝑌) ∈ V
55		fvex 6889	. . . . . . . . . . 11 ⊢ (2nd ‘𝑌) ∈ V
56	55	tposex 8259	. . . . . . . . . 10 ⊢ tpos (2nd ‘𝑌) ∈ V
57	54, 56	op1st 7996	. . . . . . . . 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 18207	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = (2nd ‘⟨𝐹, 𝑋⟩))
60		op2ndg 8001	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
61	46, 47, 60	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
62	59, 61	eqtrd 2770	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
63	58, 62	fveq12d 6883	. . . . . . 7 ⊢ (𝜑 → ((1st ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = ((1st ‘𝑌)‘𝑋))
64	53, 63	eqtrd 2770	. . . . . 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 18204	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = (1st ‘⟨𝐹, 𝑋⟩))
66		op1stg 8000	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
67	46, 47, 66	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
68	65, 67	eqtrd 2770	. . . . . 6 ⊢ (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
69	64, 68	opeq12d 4857	. . . . 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 2770	. . . 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 6880	. . 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 7408	. . 3 ⊢ (((1st ‘𝑌)‘𝑋)(1st ‘𝐻)𝐹) = ((1st ‘𝐻)‘⟨((1st ‘𝑌)‘𝑋), 𝐹⟩)
73	71, 72	eqtr4di 2788	. 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 2735	. . . 4 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
75	7, 74	fuchom 17977	. . 3 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
76	30, 10, 15, 47, 9, 22, 21, 31	yon1cl 18275	. . 3 ⊢ (𝜑 → ((1st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
77	42, 25, 8, 75, 76, 46	hof1 18266	. 2 ⊢ (𝜑 → (((1st ‘𝑌)‘𝑋)(1st ‘𝐻)𝐹) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
78	50, 73, 77	3eqtrd 2774	1 ⊢ (𝜑 → (𝐹(1st ‘𝑍)𝑋) = (((1st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ∪ cun 3924   ⊆ wss 3926  ⟨cop 4607   class class class wbr 5119   × cxp 5652  ran crn 5655  Rel wrel 5659  ‘cfv 6531  (class class class)co 7405  1^st c1st 7986  2^nd c2nd 7987  tpos ctpos 8224  Basecbs 17228  Hom chom 17282  Catccat 17676  Idccid 17677  Hom_f chomf 17678  oppCatcoppc 17723   Func cfunc 17867   ∘_func ccofu 17869   Nat cnat 17957   FuncCat cfuc 17958  SetCatcsetc 18088   ×_c cxpc 18180   1^st_F c1stf 18181   2^nd_F c2ndf 18182   ⟨,⟩_F cprf 18183   eval_F cevlf 18221  Hom_Fchof 18260  Yoncyon 18261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-map 8842  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-hom 17295  df-cco 17296  df-cat 17680  df-cid 17681  df-homf 17682  df-comf 17683  df-oppc 17724  df-func 17871  df-cofu 17873  df-nat 17959  df-fuc 17960  df-setc 18089  df-xpc 18184  df-1stf 18185  df-2ndf 18186  df-prf 18187  df-curf 18226  df-hof 18262  df-yon 18263

This theorem is referenced by:  yonedalem3a  18286  yonedalem3b  18291  yonedainv  18293  yonffthlem  18294