Theorem yonedalem3 18186

Description: Lemma for yoneda 18189. (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)
yoneda.m	⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))))

Assertion

Ref	Expression
yonedalem3	⊢ (𝜑 → 𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸))

Distinct variable groups:   𝑓,𝑎,𝑥, 1   𝐶,𝑎,𝑓,𝑥   𝐸,𝑎,𝑓   𝐵,𝑎,𝑓,𝑥   𝑂,𝑎,𝑓,𝑥   𝑆,𝑎,𝑓,𝑥   𝑄,𝑎,𝑓,𝑥   𝑇,𝑓   𝜑,𝑎,𝑓,𝑥   𝑌,𝑎,𝑓,𝑥   𝑍,𝑎,𝑓,𝑥

Allowed substitution hints:   𝑅(𝑥,𝑓,𝑎)   𝑇(𝑥,𝑎)   𝑈(𝑥,𝑓,𝑎)   𝐸(𝑥)   𝐻(𝑥,𝑓,𝑎)   𝑀(𝑥,𝑓,𝑎)   𝑉(𝑥,𝑓,𝑎)   𝑊(𝑥,𝑓,𝑎)

Proof of Theorem yonedalem3

Dummy variables 𝑔 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		yoneda.m	. . . . 5 ⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))))
2		ovex 7379	. . . . . 6 ⊢ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ∈ V
3	2	mptex 7157	. . . . 5 ⊢ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))) ∈ V
4	1, 3	fnmpoi 8002	. . . 4 ⊢ 𝑀 Fn ((𝑂 Func 𝑆) × 𝐵)
5	4	a1i 11	. . 3 ⊢ (𝜑 → 𝑀 Fn ((𝑂 Func 𝑆) × 𝐵))
6		yoneda.y	. . . . . . . 8 ⊢ 𝑌 = (Yon‘𝐶)
7		yoneda.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
8		yoneda.1	. . . . . . . 8 ⊢ 1 = (Id‘𝐶)
9		yoneda.o	. . . . . . . 8 ⊢ 𝑂 = (oppCat‘𝐶)
10		yoneda.s	. . . . . . . 8 ⊢ 𝑆 = (SetCat‘𝑈)
11		yoneda.t	. . . . . . . 8 ⊢ 𝑇 = (SetCat‘𝑉)
12		yoneda.q	. . . . . . . 8 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
13		yoneda.h	. . . . . . . 8 ⊢ 𝐻 = (HomF‘𝑄)
14		yoneda.r	. . . . . . . 8 ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
15		yoneda.e	. . . . . . . 8 ⊢ 𝐸 = (𝑂 evalF 𝑆)
16		yoneda.z	. . . . . . . 8 ⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
17		yoneda.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat)
18	17	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ Cat)
19		yoneda.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑉 ∈ 𝑊)
21		yoneda.u	. . . . . . . . 9 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
22	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → ran (Homf ‘𝐶) ⊆ 𝑈)
23		yoneda.v	. . . . . . . . 9 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
24	23	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
25		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑔 ∈ (𝑂 Func 𝑆))
26		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
27	6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 25, 26, 1	yonedalem3a 18180	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → ((𝑔𝑀𝑦) = (𝑎 ∈ (((1st ‘𝑌)‘𝑦)(𝑂 Nat 𝑆)𝑔) ↦ ((𝑎‘𝑦)‘( 1 ‘𝑦))) ∧ (𝑔𝑀𝑦):(𝑔(1st ‘𝑍)𝑦)⟶(𝑔(1st ‘𝐸)𝑦)))
28	27	simprd 495	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔𝑀𝑦):(𝑔(1st ‘𝑍)𝑦)⟶(𝑔(1st ‘𝐸)𝑦))
29		eqid 2731	. . . . . . 7 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
30		eqid 2731	. . . . . . . . . . 11 ⊢ (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
31	12	fucbas 17870	. . . . . . . . . . 11 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
32	9, 7	oppcbas 17624	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑂)
33	30, 31, 32	xpcbas 18084	. . . . . . . . . 10 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
34		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘𝑇) = (Base‘𝑇)
35		relfunc 17769	. . . . . . . . . . 11 ⊢ Rel ((𝑄 ×c 𝑂) Func 𝑇)
36	6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23	yonedalem1 18178	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
37	36	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
38		1st2ndbr 7974	. . . . . . . . . . 11 ⊢ ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st ‘𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd ‘𝑍))
39	35, 37, 38	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd ‘𝑍))
40	33, 34, 39	funcf1 17773	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
41	40	fovcdmda 7517	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1st ‘𝑍)𝑦) ∈ (Base‘𝑇))
42	11, 20	setcbas 17985	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑉 = (Base‘𝑇))
43	41, 42	eleqtrrd 2834	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1st ‘𝑍)𝑦) ∈ 𝑉)
44	36	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
45		1st2ndbr 7974	. . . . . . . . . . 11 ⊢ ((Rel ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) → (1st ‘𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd ‘𝐸))
46	35, 44, 45	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd ‘𝐸))
47	33, 34, 46	funcf1 17773	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
48	47	fovcdmda 7517	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1st ‘𝐸)𝑦) ∈ (Base‘𝑇))
49	48, 42	eleqtrrd 2834	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1st ‘𝐸)𝑦) ∈ 𝑉)
50	11, 20, 29, 43, 49	elsetchom 17988	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → ((𝑔𝑀𝑦) ∈ ((𝑔(1st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st ‘𝐸)𝑦)) ↔ (𝑔𝑀𝑦):(𝑔(1st ‘𝑍)𝑦)⟶(𝑔(1st ‘𝐸)𝑦)))
51	28, 50	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔𝑀𝑦) ∈ ((𝑔(1st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st ‘𝐸)𝑦)))
52	51	ralrimivva 3175	. . . 4 ⊢ (𝜑 → ∀𝑔 ∈ (𝑂 Func 𝑆)∀𝑦 ∈ 𝐵 (𝑔𝑀𝑦) ∈ ((𝑔(1st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st ‘𝐸)𝑦)))
53		fveq2 6822	. . . . . . 7 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀‘𝑧) = (𝑀‘⟨𝑔, 𝑦⟩))
54		df-ov 7349	. . . . . . 7 ⊢ (𝑔𝑀𝑦) = (𝑀‘⟨𝑔, 𝑦⟩)
55	53, 54	eqtr4di 2784	. . . . . 6 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀‘𝑧) = (𝑔𝑀𝑦))
56		fveq2 6822	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st ‘𝑍)‘𝑧) = ((1st ‘𝑍)‘⟨𝑔, 𝑦⟩))
57		df-ov 7349	. . . . . . . 8 ⊢ (𝑔(1st ‘𝑍)𝑦) = ((1st ‘𝑍)‘⟨𝑔, 𝑦⟩)
58	56, 57	eqtr4di 2784	. . . . . . 7 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st ‘𝑍)‘𝑧) = (𝑔(1st ‘𝑍)𝑦))
59		fveq2 6822	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st ‘𝐸)‘𝑧) = ((1st ‘𝐸)‘⟨𝑔, 𝑦⟩))
60		df-ov 7349	. . . . . . . 8 ⊢ (𝑔(1st ‘𝐸)𝑦) = ((1st ‘𝐸)‘⟨𝑔, 𝑦⟩)
61	59, 60	eqtr4di 2784	. . . . . . 7 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st ‘𝐸)‘𝑧) = (𝑔(1st ‘𝐸)𝑦))
62	58, 61	oveq12d 7364	. . . . . 6 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → (((1st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1st ‘𝐸)‘𝑧)) = ((𝑔(1st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st ‘𝐸)𝑦)))
63	55, 62	eleq12d 2825	. . . . 5 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((𝑀‘𝑧) ∈ (((1st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1st ‘𝐸)‘𝑧)) ↔ (𝑔𝑀𝑦) ∈ ((𝑔(1st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st ‘𝐸)𝑦))))
64	63	ralxp 5780	. . . 4 ⊢ (∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀‘𝑧) ∈ (((1st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1st ‘𝐸)‘𝑧)) ↔ ∀𝑔 ∈ (𝑂 Func 𝑆)∀𝑦 ∈ 𝐵 (𝑔𝑀𝑦) ∈ ((𝑔(1st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st ‘𝐸)𝑦)))
65	52, 64	sylibr 234	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀‘𝑧) ∈ (((1st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1st ‘𝐸)‘𝑧)))
66		ovex 7379	. . . . . 6 ⊢ (𝑂 Func 𝑆) ∈ V
67	7	fvexi 6836	. . . . . 6 ⊢ 𝐵 ∈ V
68	66, 67	mpoex 8011	. . . . 5 ⊢ (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥)))) ∈ V
69	1, 68	eqeltri 2827	. . . 4 ⊢ 𝑀 ∈ V
70	69	elixp 8828	. . 3 ⊢ (𝑀 ∈ X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1st ‘𝐸)‘𝑧)) ↔ (𝑀 Fn ((𝑂 Func 𝑆) × 𝐵) ∧ ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀‘𝑧) ∈ (((1st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1st ‘𝐸)‘𝑧))))
71	5, 65, 70	sylanbrc 583	. 2 ⊢ (𝜑 → 𝑀 ∈ X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1st ‘𝐸)‘𝑧)))
72	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝐶 ∈ Cat)
73	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑉 ∈ 𝑊)
74	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ran (Homf ‘𝐶) ⊆ 𝑈)
75	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
76		simpr1 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵))
77		xp1st 7953	. . . . . 6 ⊢ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → (1st ‘𝑧) ∈ (𝑂 Func 𝑆))
78	76, 77	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (1st ‘𝑧) ∈ (𝑂 Func 𝑆))
79		xp2nd 7954	. . . . . 6 ⊢ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → (2nd ‘𝑧) ∈ 𝐵)
80	76, 79	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (2nd ‘𝑧) ∈ 𝐵)
81		simpr2 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵))
82		xp1st 7953	. . . . . 6 ⊢ (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → (1st ‘𝑤) ∈ (𝑂 Func 𝑆))
83	81, 82	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (1st ‘𝑤) ∈ (𝑂 Func 𝑆))
84		xp2nd 7954	. . . . . 6 ⊢ (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → (2nd ‘𝑤) ∈ 𝐵)
85	81, 84	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (2nd ‘𝑤) ∈ 𝐵)
86		simpr3 1197	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))
87		eqid 2731	. . . . . . . . . 10 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
88	12, 87	fuchom 17871	. . . . . . . . 9 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
89		eqid 2731	. . . . . . . . 9 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
90		eqid 2731	. . . . . . . . 9 ⊢ (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
91	30, 33, 88, 89, 90, 76, 81	xpchom 18086	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤) = (((1st ‘𝑧)(𝑂 Nat 𝑆)(1st ‘𝑤)) × ((2nd ‘𝑧)(Hom ‘𝑂)(2nd ‘𝑤))))
92		eqid 2731	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
93	92, 9	oppchom 17621	. . . . . . . . 9 ⊢ ((2nd ‘𝑧)(Hom ‘𝑂)(2nd ‘𝑤)) = ((2nd ‘𝑤)(Hom ‘𝐶)(2nd ‘𝑧))
94	93	xpeq2i 5641	. . . . . . . 8 ⊢ (((1st ‘𝑧)(𝑂 Nat 𝑆)(1st ‘𝑤)) × ((2nd ‘𝑧)(Hom ‘𝑂)(2nd ‘𝑤))) = (((1st ‘𝑧)(𝑂 Nat 𝑆)(1st ‘𝑤)) × ((2nd ‘𝑤)(Hom ‘𝐶)(2nd ‘𝑧)))
95	91, 94	eqtrdi 2782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤) = (((1st ‘𝑧)(𝑂 Nat 𝑆)(1st ‘𝑤)) × ((2nd ‘𝑤)(Hom ‘𝐶)(2nd ‘𝑧))))
96	86, 95	eleqtrd 2833	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑔 ∈ (((1st ‘𝑧)(𝑂 Nat 𝑆)(1st ‘𝑤)) × ((2nd ‘𝑤)(Hom ‘𝐶)(2nd ‘𝑧))))
97		xp1st 7953	. . . . . 6 ⊢ (𝑔 ∈ (((1st ‘𝑧)(𝑂 Nat 𝑆)(1st ‘𝑤)) × ((2nd ‘𝑤)(Hom ‘𝐶)(2nd ‘𝑧))) → (1st ‘𝑔) ∈ ((1st ‘𝑧)(𝑂 Nat 𝑆)(1st ‘𝑤)))
98	96, 97	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (1st ‘𝑔) ∈ ((1st ‘𝑧)(𝑂 Nat 𝑆)(1st ‘𝑤)))
99		xp2nd 7954	. . . . . 6 ⊢ (𝑔 ∈ (((1st ‘𝑧)(𝑂 Nat 𝑆)(1st ‘𝑤)) × ((2nd ‘𝑤)(Hom ‘𝐶)(2nd ‘𝑧))) → (2nd ‘𝑔) ∈ ((2nd ‘𝑤)(Hom ‘𝐶)(2nd ‘𝑧)))
100	96, 99	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (2nd ‘𝑔) ∈ ((2nd ‘𝑤)(Hom ‘𝐶)(2nd ‘𝑧)))
101	6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 72, 73, 74, 75, 78, 80, 83, 85, 98, 100, 1	yonedalem3b 18185	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (((1st ‘𝑤)𝑀(2nd ‘𝑤))(⟨((1st ‘𝑧)(1st ‘𝑍)(2nd ‘𝑧)), ((1st ‘𝑤)(1st ‘𝑍)(2nd ‘𝑤))⟩(comp‘𝑇)((1st ‘𝑤)(1st ‘𝐸)(2nd ‘𝑤)))((1st ‘𝑔)(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝑍)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)(2nd ‘𝑔))) = (((1st ‘𝑔)(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝐸)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)(2nd ‘𝑔))(⟨((1st ‘𝑧)(1st ‘𝑍)(2nd ‘𝑧)), ((1st ‘𝑧)(1st ‘𝐸)(2nd ‘𝑧))⟩(comp‘𝑇)((1st ‘𝑤)(1st ‘𝐸)(2nd ‘𝑤)))((1st ‘𝑧)𝑀(2nd ‘𝑧))))
102		1st2nd2 7960	. . . . . . . . . 10 ⊢ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
103	76, 102	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
104	103	fveq2d 6826	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st ‘𝑍)‘𝑧) = ((1st ‘𝑍)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
105		df-ov 7349	. . . . . . . 8 ⊢ ((1st ‘𝑧)(1st ‘𝑍)(2nd ‘𝑧)) = ((1st ‘𝑍)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
106	104, 105	eqtr4di 2784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st ‘𝑍)‘𝑧) = ((1st ‘𝑧)(1st ‘𝑍)(2nd ‘𝑧)))
107		1st2nd2 7960	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
108	81, 107	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
109	108	fveq2d 6826	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st ‘𝑍)‘𝑤) = ((1st ‘𝑍)‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩))
110		df-ov 7349	. . . . . . . 8 ⊢ ((1st ‘𝑤)(1st ‘𝑍)(2nd ‘𝑤)) = ((1st ‘𝑍)‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
111	109, 110	eqtr4di 2784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st ‘𝑍)‘𝑤) = ((1st ‘𝑤)(1st ‘𝑍)(2nd ‘𝑤)))
112	106, 111	opeq12d 4830	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ⟨((1st ‘𝑍)‘𝑧), ((1st ‘𝑍)‘𝑤)⟩ = ⟨((1st ‘𝑧)(1st ‘𝑍)(2nd ‘𝑧)), ((1st ‘𝑤)(1st ‘𝑍)(2nd ‘𝑤))⟩)
113	108	fveq2d 6826	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st ‘𝐸)‘𝑤) = ((1st ‘𝐸)‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩))
114		df-ov 7349	. . . . . . 7 ⊢ ((1st ‘𝑤)(1st ‘𝐸)(2nd ‘𝑤)) = ((1st ‘𝐸)‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
115	113, 114	eqtr4di 2784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st ‘𝐸)‘𝑤) = ((1st ‘𝑤)(1st ‘𝐸)(2nd ‘𝑤)))
116	112, 115	oveq12d 7364	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (⟨((1st ‘𝑍)‘𝑧), ((1st ‘𝑍)‘𝑤)⟩(comp‘𝑇)((1st ‘𝐸)‘𝑤)) = (⟨((1st ‘𝑧)(1st ‘𝑍)(2nd ‘𝑧)), ((1st ‘𝑤)(1st ‘𝑍)(2nd ‘𝑤))⟩(comp‘𝑇)((1st ‘𝑤)(1st ‘𝐸)(2nd ‘𝑤))))
117	108	fveq2d 6826	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀‘𝑤) = (𝑀‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩))
118		df-ov 7349	. . . . . 6 ⊢ ((1st ‘𝑤)𝑀(2nd ‘𝑤)) = (𝑀‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
119	117, 118	eqtr4di 2784	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀‘𝑤) = ((1st ‘𝑤)𝑀(2nd ‘𝑤)))
120	103, 108	oveq12d 7364	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑧(2nd ‘𝑍)𝑤) = (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝑍)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩))
121		1st2nd2 7960	. . . . . . . 8 ⊢ (𝑔 ∈ (((1st ‘𝑧)(𝑂 Nat 𝑆)(1st ‘𝑤)) × ((2nd ‘𝑤)(Hom ‘𝐶)(2nd ‘𝑧))) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
122	96, 121	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑔 = ⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
123	120, 122	fveq12d 6829	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑧(2nd ‘𝑍)𝑤)‘𝑔) = ((⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝑍)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)‘⟨(1st ‘𝑔), (2nd ‘𝑔)⟩))
124		df-ov 7349	. . . . . 6 ⊢ ((1st ‘𝑔)(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝑍)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)(2nd ‘𝑔)) = ((⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝑍)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)‘⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
125	123, 124	eqtr4di 2784	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑧(2nd ‘𝑍)𝑤)‘𝑔) = ((1st ‘𝑔)(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝑍)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)(2nd ‘𝑔)))
126	116, 119, 125	oveq123d 7367	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑀‘𝑤)(⟨((1st ‘𝑍)‘𝑧), ((1st ‘𝑍)‘𝑤)⟩(comp‘𝑇)((1st ‘𝐸)‘𝑤))((𝑧(2nd ‘𝑍)𝑤)‘𝑔)) = (((1st ‘𝑤)𝑀(2nd ‘𝑤))(⟨((1st ‘𝑧)(1st ‘𝑍)(2nd ‘𝑧)), ((1st ‘𝑤)(1st ‘𝑍)(2nd ‘𝑤))⟩(comp‘𝑇)((1st ‘𝑤)(1st ‘𝐸)(2nd ‘𝑤)))((1st ‘𝑔)(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝑍)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)(2nd ‘𝑔))))
127	103	fveq2d 6826	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st ‘𝐸)‘𝑧) = ((1st ‘𝐸)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
128		df-ov 7349	. . . . . . . 8 ⊢ ((1st ‘𝑧)(1st ‘𝐸)(2nd ‘𝑧)) = ((1st ‘𝐸)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
129	127, 128	eqtr4di 2784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((1st ‘𝐸)‘𝑧) = ((1st ‘𝑧)(1st ‘𝐸)(2nd ‘𝑧)))
130	106, 129	opeq12d 4830	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ⟨((1st ‘𝑍)‘𝑧), ((1st ‘𝐸)‘𝑧)⟩ = ⟨((1st ‘𝑧)(1st ‘𝑍)(2nd ‘𝑧)), ((1st ‘𝑧)(1st ‘𝐸)(2nd ‘𝑧))⟩)
131	130, 115	oveq12d 7364	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (⟨((1st ‘𝑍)‘𝑧), ((1st ‘𝐸)‘𝑧)⟩(comp‘𝑇)((1st ‘𝐸)‘𝑤)) = (⟨((1st ‘𝑧)(1st ‘𝑍)(2nd ‘𝑧)), ((1st ‘𝑧)(1st ‘𝐸)(2nd ‘𝑧))⟩(comp‘𝑇)((1st ‘𝑤)(1st ‘𝐸)(2nd ‘𝑤))))
132	103, 108	oveq12d 7364	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑧(2nd ‘𝐸)𝑤) = (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝐸)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩))
133	132, 122	fveq12d 6829	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑧(2nd ‘𝐸)𝑤)‘𝑔) = ((⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝐸)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)‘⟨(1st ‘𝑔), (2nd ‘𝑔)⟩))
134		df-ov 7349	. . . . . 6 ⊢ ((1st ‘𝑔)(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝐸)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)(2nd ‘𝑔)) = ((⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝐸)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)‘⟨(1st ‘𝑔), (2nd ‘𝑔)⟩)
135	133, 134	eqtr4di 2784	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑧(2nd ‘𝐸)𝑤)‘𝑔) = ((1st ‘𝑔)(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝐸)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)(2nd ‘𝑔)))
136	103	fveq2d 6826	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀‘𝑧) = (𝑀‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
137		df-ov 7349	. . . . . 6 ⊢ ((1st ‘𝑧)𝑀(2nd ‘𝑧)) = (𝑀‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
138	136, 137	eqtr4di 2784	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (𝑀‘𝑧) = ((1st ‘𝑧)𝑀(2nd ‘𝑧)))
139	131, 135, 138	oveq123d 7367	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → (((𝑧(2nd ‘𝐸)𝑤)‘𝑔)(⟨((1st ‘𝑍)‘𝑧), ((1st ‘𝐸)‘𝑧)⟩(comp‘𝑇)((1st ‘𝐸)‘𝑤))(𝑀‘𝑧)) = (((1st ‘𝑔)(⟨(1st ‘𝑧), (2nd ‘𝑧)⟩(2nd ‘𝐸)⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)(2nd ‘𝑔))(⟨((1st ‘𝑧)(1st ‘𝑍)(2nd ‘𝑧)), ((1st ‘𝑧)(1st ‘𝐸)(2nd ‘𝑧))⟩(comp‘𝑇)((1st ‘𝑤)(1st ‘𝐸)(2nd ‘𝑤)))((1st ‘𝑧)𝑀(2nd ‘𝑧))))
140	101, 126, 139	3eqtr4d 2776	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → ((𝑀‘𝑤)(⟨((1st ‘𝑍)‘𝑧), ((1st ‘𝑍)‘𝑤)⟩(comp‘𝑇)((1st ‘𝐸)‘𝑤))((𝑧(2nd ‘𝑍)𝑤)‘𝑔)) = (((𝑧(2nd ‘𝐸)𝑤)‘𝑔)(⟨((1st ‘𝑍)‘𝑧), ((1st ‘𝐸)‘𝑧)⟩(comp‘𝑇)((1st ‘𝐸)‘𝑤))(𝑀‘𝑧)))
141	140	ralrimivvva 3178	. 2 ⊢ (𝜑 → ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤)((𝑀‘𝑤)(⟨((1st ‘𝑍)‘𝑧), ((1st ‘𝑍)‘𝑤)⟩(comp‘𝑇)((1st ‘𝐸)‘𝑤))((𝑧(2nd ‘𝑍)𝑤)‘𝑔)) = (((𝑧(2nd ‘𝐸)𝑤)‘𝑔)(⟨((1st ‘𝑍)‘𝑧), ((1st ‘𝐸)‘𝑧)⟩(comp‘𝑇)((1st ‘𝐸)‘𝑤))(𝑀‘𝑧)))
142		eqid 2731	. . 3 ⊢ ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
143		eqid 2731	. . 3 ⊢ (comp‘𝑇) = (comp‘𝑇)
144	142, 33, 90, 29, 143, 37, 44	isnat2 17858	. 2 ⊢ (𝜑 → (𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸) ↔ (𝑀 ∈ X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1st ‘𝐸)‘𝑧)) ∧ ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤)((𝑀‘𝑤)(⟨((1st ‘𝑍)‘𝑧), ((1st ‘𝑍)‘𝑤)⟩(comp‘𝑇)((1st ‘𝐸)‘𝑤))((𝑧(2nd ‘𝑍)𝑤)‘𝑔)) = (((𝑧(2nd ‘𝐸)𝑤)‘𝑔)(⟨((1st ‘𝑍)‘𝑧), ((1st ‘𝐸)‘𝑧)⟩(comp‘𝑇)((1st ‘𝐸)‘𝑤))(𝑀‘𝑧)))))
145	71, 141, 144	mpbir2and 713	1 ⊢ (𝜑 → 𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ∪ cun 3895   ⊆ wss 3897  ⟨cop 4579   class class class wbr 5089   ↦ cmpt 5170   × cxp 5612  ran crn 5615  Rel wrel 5619   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920  tpos ctpos 8155  Xcixp 8821  Basecbs 17120  Hom chom 17172  compcco 17173  Catccat 17570  Idccid 17571  Hom_f chomf 17572  oppCatcoppc 17617   Func cfunc 17761   ∘_func ccofu 17763   Nat cnat 17851   FuncCat cfuc 17852  SetCatcsetc 17982   ×_c cxpc 18074   1^st_F c1stf 18075   2^nd_F c2ndf 18076   ⟨,⟩_F cprf 18077   eval_F cevlf 18115  Hom_Fchof 18154  Yoncyon 18155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-tpos 8156  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-pm 8753  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-uz 12733  df-fz 13408  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-hom 17185  df-cco 17186  df-cat 17574  df-cid 17575  df-homf 17576  df-comf 17577  df-oppc 17618  df-ssc 17717  df-resc 17718  df-subc 17719  df-func 17765  df-cofu 17767  df-nat 17853  df-fuc 17854  df-setc 17983  df-xpc 18078  df-1stf 18079  df-2ndf 18080  df-prf 18081  df-evlf 18119  df-curf 18120  df-hof 18156  df-yon 18157

This theorem is referenced by:  yonedainv  18187