Theorem yonedalem22 18203

Description: Lemma for yoneda 18208. (Contributed by Mario Carneiro, 29-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)
yonedalem22.g	⊢ (𝜑 → 𝐺 ∈ (𝑂 Func 𝑆))
yonedalem22.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
yonedalem22.a	⊢ (𝜑 → 𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
yonedalem22.k	⊢ (𝜑 → 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))

Assertion

Ref	Expression
yonedalem22	⊢ (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd ‘𝑍)⟨𝐺, 𝑃⟩)𝐾) = (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)(⟨((1st ‘𝑌)‘𝑋), 𝐹⟩(2nd ‘𝐻)⟨((1st ‘𝑌)‘𝑃), 𝐺⟩)𝐴))

Proof of Theorem yonedalem22

Step	Hyp	Ref	Expression
1		yoneda.z	. . . . . . 7 ⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
2	1	fveq2i 6836	. . . . . 6 ⊢ (2nd ‘𝑍) = (2nd ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))
3	2	oveqi 7371	. . . . 5 ⊢ (⟨𝐹, 𝑋⟩(2nd ‘𝑍)⟨𝐺, 𝑃⟩) = (⟨𝐹, 𝑋⟩(2nd ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)
4	3	oveqi 7371	. . . 4 ⊢ (𝐴(⟨𝐹, 𝑋⟩(2nd ‘𝑍)⟨𝐺, 𝑃⟩)𝐾) = (𝐴(⟨𝐹, 𝑋⟩(2nd ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)𝐾)
5		df-ov 7361	. . . 4 ⊢ (𝐴(⟨𝐹, 𝑋⟩(2nd ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)𝐾) = ((⟨𝐹, 𝑋⟩(2nd ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)
6	4, 5	eqtri 2758	. . 3 ⊢ (𝐴(⟨𝐹, 𝑋⟩(2nd ‘𝑍)⟨𝐺, 𝑃⟩)𝐾) = ((⟨𝐹, 𝑋⟩(2nd ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)
7		eqid 2735	. . . . 5 ⊢ (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
8		yoneda.q	. . . . . 6 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
9	8	fucbas 17889	. . . . 5 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
10		yoneda.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
11		yoneda.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
12	10, 11	oppcbas 17643	. . . . 5 ⊢ 𝐵 = (Base‘𝑂)
13	7, 9, 12	xpcbas 18103	. . . 4 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
14		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))
15		eqid 2735	. . . . 5 ⊢ ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄)
16		yoneda.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
17	10	oppccat 17647	. . . . . . . . 9 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
18	16, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ Cat)
19		yoneda.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
20		yoneda.v	. . . . . . . . . . 11 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
21	20	unssbd 4145	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
22	19, 21	ssexd 5268	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V)
23		yoneda.s	. . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈)
24	23	setccat 18011	. . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
25	22, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat)
26	8, 18, 25	fuccat 17899	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat)
27		eqid 2735	. . . . . . 7 ⊢ (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂)
28	7, 26, 18, 27	2ndfcl 18123	. . . . . 6 ⊢ (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂))
29		eqid 2735	. . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄)
30		relfunc 17788	. . . . . . . . 9 ⊢ Rel (𝐶 Func 𝑄)
31		yoneda.y	. . . . . . . . . 10 ⊢ 𝑌 = (Yon‘𝐶)
32		yoneda.u	. . . . . . . . . 10 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
33	31, 16, 10, 23, 8, 22, 32	yoncl 18187	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
34		1st2ndbr 7986	. . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
35	30, 33, 34	sylancr 588	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
36	10, 29, 35	funcoppc 17801	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌))
37		df-br 5098	. . . . . . 7 ⊢ ((1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌) ↔ ⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
38	36, 37	sylib 218	. . . . . 6 ⊢ (𝜑 → ⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∈ (𝑂 Func (oppCat‘𝑄)))
39	28, 38	cofucl 17814	. . . . 5 ⊢ (𝜑 → (⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄)))
40		eqid 2735	. . . . . 6 ⊢ (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂)
41	7, 26, 18, 40	1stfcl 18122	. . . . 5 ⊢ (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄))
42	14, 15, 39, 41	prfcl 18128	. . . 4 ⊢ (𝜑 → ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func ((oppCat‘𝑄) ×c 𝑄)))
43		yoneda.h	. . . . 5 ⊢ 𝐻 = (HomF‘𝑄)
44		yoneda.t	. . . . 5 ⊢ 𝑇 = (SetCat‘𝑉)
45	20	unssad 4144	. . . . 5 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉)
46	43, 29, 44, 26, 19, 45	hofcl 18184	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇))
47		yonedalem21.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))
48		yonedalem21.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
49	47, 48	opelxpd 5662	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝑋⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
50		yonedalem22.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝑂 Func 𝑆))
51		yonedalem22.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
52	50, 51	opelxpd 5662	. . . 4 ⊢ (𝜑 → ⟨𝐺, 𝑃⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
53		eqid 2735	. . . 4 ⊢ (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
54		yonedalem22.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺))
55		yonedalem22.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋))
56		eqid 2735	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
57	56, 10	oppchom 17640	. . . . . . 7 ⊢ (𝑋(Hom ‘𝑂)𝑃) = (𝑃(Hom ‘𝐶)𝑋)
58	55, 57	eleqtrrdi 2846	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑋(Hom ‘𝑂)𝑃))
59	54, 58	opelxpd 5662	. . . . 5 ⊢ (𝜑 → ⟨𝐴, 𝐾⟩ ∈ ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
60		eqid 2735	. . . . . . 7 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
61	8, 60	fuchom 17890	. . . . . 6 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
62		eqid 2735	. . . . . 6 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
63	7, 9, 12, 61, 62, 47, 48, 50, 51, 53	xpchom2 18111	. . . . 5 ⊢ (𝜑 → (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩) = ((𝐹(𝑂 Nat 𝑆)𝐺) × (𝑋(Hom ‘𝑂)𝑃)))
64	59, 63	eleqtrrd 2838	. . . 4 ⊢ (𝜑 → ⟨𝐴, 𝐾⟩ ∈ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))
65	13, 42, 46, 49, 52, 53, 64	cofu2 17812	. . 3 ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂))))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)(2nd ‘𝐻)((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩))‘((⟨𝐹, 𝑋⟩(2nd ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)))
66	6, 65	eqtrid 2782	. 2 ⊢ (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd ‘𝑍)⟨𝐺, 𝑃⟩)𝐾) = ((((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)(2nd ‘𝐻)((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩))‘((⟨𝐹, 𝑋⟩(2nd ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)))
67	14, 13, 53, 39, 41, 49	prf1 18125	. . . . . 6 ⊢ (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⟩)⟩)
68	13, 28, 38, 49	cofu1 17810	. . . . . . . 8 ⊢ (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⟩)))
69		fvex 6846	. . . . . . . . . . 11 ⊢ (1st ‘𝑌) ∈ V
70		fvex 6846	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑌) ∈ V
71	70	tposex 8202	. . . . . . . . . . 11 ⊢ tpos (2nd ‘𝑌) ∈ V
72	69, 71	op1st 7941	. . . . . . . . . 10 ⊢ (1st ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩) = (1st ‘𝑌)
73	72	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩) = (1st ‘𝑌))
74	7, 13, 53, 26, 18, 27, 49	2ndf1 18120	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = (2nd ‘⟨𝐹, 𝑋⟩))
75		op2ndg 7946	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐵) → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
76	47, 48, 75	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
77	74, 76	eqtrd 2770	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝑋)
78	73, 77	fveq12d 6840	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = ((1st ‘𝑌)‘𝑋))
79	68, 78	eqtrd 2770	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩) = ((1st ‘𝑌)‘𝑋))
80	7, 13, 53, 26, 18, 40, 49	1stf1 18117	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = (1st ‘⟨𝐹, 𝑋⟩))
81		op1stg 7945	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑂 Func 𝑆) ∧ 𝑋 ∈ 𝐵) → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
82	47, 48, 81	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝑋⟩) = 𝐹)
83	80, 82	eqtrd 2770	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩) = 𝐹)
84	79, 83	opeq12d 4836	. . . . . 6 ⊢ (𝜑 → ⟨((1st ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐹, 𝑋⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐹, 𝑋⟩)⟩ = ⟨((1st ‘𝑌)‘𝑋), 𝐹⟩)
85	67, 84	eqtrd 2770	. . . . 5 ⊢ (𝜑 → ((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩) = ⟨((1st ‘𝑌)‘𝑋), 𝐹⟩)
86	14, 13, 53, 39, 41, 52	prf1 18125	. . . . . 6 ⊢ (𝜑 → ((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩) = ⟨((1st ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩)⟩)
87	13, 28, 38, 52	cofu1 17810	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩) = ((1st ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)))
88	7, 13, 53, 26, 18, 27, 52	2ndf1 18120	. . . . . . . . . 10 ⊢ (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩) = (2nd ‘⟨𝐺, 𝑃⟩))
89		op2ndg 7946	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃 ∈ 𝐵) → (2nd ‘⟨𝐺, 𝑃⟩) = 𝑃)
90	50, 51, 89	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘⟨𝐺, 𝑃⟩) = 𝑃)
91	88, 90	eqtrd 2770	. . . . . . . . 9 ⊢ (𝜑 → ((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩) = 𝑃)
92	73, 91	fveq12d 6840	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩)‘((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)) = ((1st ‘𝑌)‘𝑃))
93	87, 92	eqtrd 2770	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩) = ((1st ‘𝑌)‘𝑃))
94	7, 13, 53, 26, 18, 40, 52	1stf1 18117	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩) = (1st ‘⟨𝐺, 𝑃⟩))
95		op1stg 7945	. . . . . . . . 9 ⊢ ((𝐺 ∈ (𝑂 Func 𝑆) ∧ 𝑃 ∈ 𝐵) → (1st ‘⟨𝐺, 𝑃⟩) = 𝐺)
96	50, 51, 95	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝐺, 𝑃⟩) = 𝐺)
97	94, 96	eqtrd 2770	. . . . . . 7 ⊢ (𝜑 → ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩) = 𝐺)
98	93, 97	opeq12d 4836	. . . . . 6 ⊢ (𝜑 → ⟨((1st ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))‘⟨𝐺, 𝑃⟩), ((1st ‘(𝑄 1stF 𝑂))‘⟨𝐺, 𝑃⟩)⟩ = ⟨((1st ‘𝑌)‘𝑃), 𝐺⟩)
99	86, 98	eqtrd 2770	. . . . 5 ⊢ (𝜑 → ((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩) = ⟨((1st ‘𝑌)‘𝑃), 𝐺⟩)
100	85, 99	oveq12d 7376	. . . 4 ⊢ (𝜑 → (((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)(2nd ‘𝐻)((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩)) = (⟨((1st ‘𝑌)‘𝑋), 𝐹⟩(2nd ‘𝐻)⟨((1st ‘𝑌)‘𝑃), 𝐺⟩))
101	14, 13, 53, 39, 41, 49, 52, 64	prf2 18127	. . . . 5 ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ⟨((⟨𝐹, 𝑋⟩(2nd ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩), ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)⟩)
102	13, 28, 38, 49, 52, 53, 64	cofu2 17812	. . . . . . 7 ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)(2nd ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩))‘((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)))
103	69, 71	op2nd 7942	. . . . . . . . . . 11 ⊢ (2nd ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩) = tpos (2nd ‘𝑌)
104	103	oveqi 7371	. . . . . . . . . 10 ⊢ (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)(2nd ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)) = (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)tpos (2nd ‘𝑌)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩))
105		ovtpos 8183	. . . . . . . . . 10 ⊢ (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)tpos (2nd ‘𝑌)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)) = (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)(2nd ‘𝑌)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩))
106	104, 105	eqtri 2758	. . . . . . . . 9 ⊢ (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)(2nd ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)) = (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)(2nd ‘𝑌)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩))
107	91, 77	oveq12d 7376	. . . . . . . . 9 ⊢ (𝜑 → (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)(2nd ‘𝑌)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)) = (𝑃(2nd ‘𝑌)𝑋))
108	106, 107	eqtrid 2782	. . . . . . . 8 ⊢ (𝜑 → (((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)(2nd ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩)) = (𝑃(2nd ‘𝑌)𝑋))
109	7, 13, 53, 26, 18, 27, 49, 52	2ndf2 18121	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩) = (2nd ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)))
110	109	fveq1d 6835	. . . . . . . . 9 ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((2nd ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩))
111	64	fvresd 6853	. . . . . . . . 9 ⊢ (𝜑 → ((2nd ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩) = (2nd ‘⟨𝐴, 𝐾⟩))
112		op2ndg 7946	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺) ∧ 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋)) → (2nd ‘⟨𝐴, 𝐾⟩) = 𝐾)
113	54, 55, 112	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (2nd ‘⟨𝐴, 𝐾⟩) = 𝐾)
114	110, 111, 113	3eqtrd 2774	. . . . . . . 8 ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = 𝐾)
115	108, 114	fveq12d 6840	. . . . . . 7 ⊢ (𝜑 → ((((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐹, 𝑋⟩)(2nd ‘⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩)((1st ‘(𝑄 2ndF 𝑂))‘⟨𝐺, 𝑃⟩))‘((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 2ndF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)) = ((𝑃(2nd ‘𝑌)𝑋)‘𝐾))
116	102, 115	eqtrd 2770	. . . . . 6 ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((𝑃(2nd ‘𝑌)𝑋)‘𝐾))
117	7, 13, 53, 26, 18, 40, 49, 52	1stf2 18118	. . . . . . . 8 ⊢ (𝜑 → (⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩) = (1st ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩)))
118	117	fveq1d 6835	. . . . . . 7 ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ((1st ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩))
119	64	fvresd 6853	. . . . . . 7 ⊢ (𝜑 → ((1st ↾ (⟨𝐹, 𝑋⟩(Hom ‘(𝑄 ×c 𝑂))⟨𝐺, 𝑃⟩))‘⟨𝐴, 𝐾⟩) = (1st ‘⟨𝐴, 𝐾⟩))
120		op1stg 7945	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝐹(𝑂 Nat 𝑆)𝐺) ∧ 𝐾 ∈ (𝑃(Hom ‘𝐶)𝑋)) → (1st ‘⟨𝐴, 𝐾⟩) = 𝐴)
121	54, 55, 120	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (1st ‘⟨𝐴, 𝐾⟩) = 𝐴)
122	118, 119, 121	3eqtrd 2774	. . . . . 6 ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = 𝐴)
123	116, 122	opeq12d 4836	. . . . 5 ⊢ (𝜑 → ⟨((⟨𝐹, 𝑋⟩(2nd ‘(⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩), ((⟨𝐹, 𝑋⟩(2nd ‘(𝑄 1stF 𝑂))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)⟩ = ⟨((𝑃(2nd ‘𝑌)𝑋)‘𝐾), 𝐴⟩)
124	101, 123	eqtrd 2770	. . . 4 ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2nd ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩) = ⟨((𝑃(2nd ‘𝑌)𝑋)‘𝐾), 𝐴⟩)
125	100, 124	fveq12d 6840	. . 3 ⊢ (𝜑 → ((((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)(2nd ‘𝐻)((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩))‘((⟨𝐹, 𝑋⟩(2nd ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)) = ((⟨((1st ‘𝑌)‘𝑋), 𝐹⟩(2nd ‘𝐻)⟨((1st ‘𝑌)‘𝑃), 𝐺⟩)‘⟨((𝑃(2nd ‘𝑌)𝑋)‘𝐾), 𝐴⟩))
126		df-ov 7361	. . 3 ⊢ (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)(⟨((1st ‘𝑌)‘𝑋), 𝐹⟩(2nd ‘𝐻)⟨((1st ‘𝑌)‘𝑃), 𝐺⟩)𝐴) = ((⟨((1st ‘𝑌)‘𝑋), 𝐹⟩(2nd ‘𝐻)⟨((1st ‘𝑌)‘𝑃), 𝐺⟩)‘⟨((𝑃(2nd ‘𝑌)𝑋)‘𝐾), 𝐴⟩)
127	125, 126	eqtr4di 2788	. 2 ⊢ (𝜑 → ((((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐹, 𝑋⟩)(2nd ‘𝐻)((1st ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))‘⟨𝐺, 𝑃⟩))‘((⟨𝐹, 𝑋⟩(2nd ‘((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))⟨𝐺, 𝑃⟩)‘⟨𝐴, 𝐾⟩)) = (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)(⟨((1st ‘𝑌)‘𝑋), 𝐹⟩(2nd ‘𝐻)⟨((1st ‘𝑌)‘𝑃), 𝐺⟩)𝐴))
128	66, 127	eqtrd 2770	1 ⊢ (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd ‘𝑍)⟨𝐺, 𝑃⟩)𝐾) = (((𝑃(2nd ‘𝑌)𝑋)‘𝐾)(⟨((1st ‘𝑌)‘𝑋), 𝐹⟩(2nd ‘𝐻)⟨((1st ‘𝑌)‘𝑃), 𝐺⟩)𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3439   ∪ cun 3898   ⊆ wss 3900  ⟨cop 4585   class class class wbr 5097   × cxp 5621  ran crn 5624   ↾ cres 5625  Rel wrel 5628  ‘cfv 6491  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  tpos ctpos 8167  Basecbs 17138  Hom chom 17190  Catccat 17589  Idccid 17590  Hom_f chomf 17591  oppCatcoppc 17636   Func cfunc 17780   ∘_func ccofu 17782   Nat cnat 17870   FuncCat cfuc 17871  SetCatcsetc 18001   ×_c cxpc 18093   1^st_F c1stf 18094   2^nd_F c2ndf 18095   ⟨,⟩_F cprf 18096   eval_F cevlf 18134  Hom_Fchof 18173  Yoncyon 18174

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8767  df-ixp 8838  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12610  df-uz 12754  df-fz 13426  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-hom 17203  df-cco 17204  df-cat 17593  df-cid 17594  df-homf 17595  df-comf 17596  df-oppc 17637  df-func 17784  df-cofu 17786  df-nat 17872  df-fuc 17873  df-setc 18002  df-xpc 18097  df-1stf 18098  df-2ndf 18099  df-prf 18100  df-curf 18139  df-hof 18175  df-yon 18176

This theorem is referenced by:  yonedalem3b  18204