Theorem yonedalem3 17998

Description: Lemma for yoneda 18001. (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 7308	. . . . . 6 ⊢ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ∈ V
3	2	mptex 7099	. . . . 5 ⊢ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))) ∈ V
4	1, 3	fnmpoi 7910	. . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ Cat)
19		yoneda.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
20	19	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑉 ∈ 𝑊)
21		yoneda.u	. . . . . . . . 9 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
22	21	adantr 481	. . . . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
25		simprl 768	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑔 ∈ (𝑂 Func 𝑆))
26		simprr 770	. . . . . . . 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 17992	. . . . . . 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 496	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔𝑀𝑦):(𝑔(1st ‘𝑍)𝑦)⟶(𝑔(1st ‘𝐸)𝑦))
29		eqid 2738	. . . . . . 7 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
30		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
31	12	fucbas 17677	. . . . . . . . . . 11 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
32	9, 7	oppcbas 17428	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑂)
33	30, 31, 32	xpcbas 17895	. . . . . . . . . 10 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
34		eqid 2738	. . . . . . . . . 10 ⊢ (Base‘𝑇) = (Base‘𝑇)
35		relfunc 17577	. . . . . . . . . . 11 ⊢ Rel ((𝑄 ×c 𝑂) Func 𝑇)
36	6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23	yonedalem1 17990	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
37	36	simpld 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
38		1st2ndbr 7883	. . . . . . . . . . 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 17581	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
41	40	fovrnda 7443	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1st ‘𝑍)𝑦) ∈ (Base‘𝑇))
42	11, 20	setcbas 17793	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑉 = (Base‘𝑇))
43	41, 42	eleqtrrd 2842	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1st ‘𝑍)𝑦) ∈ 𝑉)
44	36	simprd 496	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
45		1st2ndbr 7883	. . . . . . . . . . 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 17581	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
48	47	fovrnda 7443	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1st ‘𝐸)𝑦) ∈ (Base‘𝑇))
49	48, 42	eleqtrrd 2842	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1st ‘𝐸)𝑦) ∈ 𝑉)
50	11, 20, 29, 43, 49	elsetchom 17796	. . . . . 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 256	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔𝑀𝑦) ∈ ((𝑔(1st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st ‘𝐸)𝑦)))
52	51	ralrimivva 3123	. . . 4 ⊢ (𝜑 → ∀𝑔 ∈ (𝑂 Func 𝑆)∀𝑦 ∈ 𝐵 (𝑔𝑀𝑦) ∈ ((𝑔(1st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st ‘𝐸)𝑦)))
53		fveq2 6774	. . . . . . 7 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀‘𝑧) = (𝑀‘⟨𝑔, 𝑦⟩))
54		df-ov 7278	. . . . . . 7 ⊢ (𝑔𝑀𝑦) = (𝑀‘⟨𝑔, 𝑦⟩)
55	53, 54	eqtr4di 2796	. . . . . 6 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀‘𝑧) = (𝑔𝑀𝑦))
56		fveq2 6774	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st ‘𝑍)‘𝑧) = ((1st ‘𝑍)‘⟨𝑔, 𝑦⟩))
57		df-ov 7278	. . . . . . . 8 ⊢ (𝑔(1st ‘𝑍)𝑦) = ((1st ‘𝑍)‘⟨𝑔, 𝑦⟩)
58	56, 57	eqtr4di 2796	. . . . . . 7 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st ‘𝑍)‘𝑧) = (𝑔(1st ‘𝑍)𝑦))
59		fveq2 6774	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st ‘𝐸)‘𝑧) = ((1st ‘𝐸)‘⟨𝑔, 𝑦⟩))
60		df-ov 7278	. . . . . . . 8 ⊢ (𝑔(1st ‘𝐸)𝑦) = ((1st ‘𝐸)‘⟨𝑔, 𝑦⟩)
61	59, 60	eqtr4di 2796	. . . . . . 7 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st ‘𝐸)‘𝑧) = (𝑔(1st ‘𝐸)𝑦))
62	58, 61	oveq12d 7293	. . . . . 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 2833	. . . . 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 5750	. . . 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 233	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀‘𝑧) ∈ (((1st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1st ‘𝐸)‘𝑧)))
66		ovex 7308	. . . . . 6 ⊢ (𝑂 Func 𝑆) ∈ V
67	7	fvexi 6788	. . . . . 6 ⊢ 𝐵 ∈ V
68	66, 67	mpoex 7920	. . . . 5 ⊢ (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥)))) ∈ V
69	1, 68	eqeltri 2835	. . . 4 ⊢ 𝑀 ∈ V
70	69	elixp 8692	. . 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 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝐶 ∈ Cat)
73	19	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×c 𝑂))𝑤))) → 𝑉 ∈ 𝑊)
74	21	adantr 481	. . . . 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 481	. . . . 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 1193	. . . . . 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 7863	. . . . . 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 7864	. . . . . 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 1194	. . . . . 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 7863	. . . . . 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 7864	. . . . . 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 1195	. . . . . . 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 2738	. . . . . . . . . 10 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
88	12, 87	fuchom 17678	. . . . . . . . 9 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
89		eqid 2738	. . . . . . . . 9 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
90		eqid 2738	. . . . . . . . 9 ⊢ (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
91	30, 33, 88, 89, 90, 76, 81	xpchom 17897	. . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
93	92, 9	oppchom 17425	. . . . . . . . 9 ⊢ ((2nd ‘𝑧)(Hom ‘𝑂)(2nd ‘𝑤)) = ((2nd ‘𝑤)(Hom ‘𝐶)(2nd ‘𝑧))
94	93	xpeq2i 5616	. . . . . . . 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 2794	. . . . . . 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 2841	. . . . . 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 7863	. . . . . 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 7864	. . . . . 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 17997	. . . 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 7870	. . . . . . . . . 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 6778	. . . . . . . 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 7278	. . . . . . . 8 ⊢ ((1st ‘𝑧)(1st ‘𝑍)(2nd ‘𝑧)) = ((1st ‘𝑍)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
106	104, 105	eqtr4di 2796	. . . . . . 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 7870	. . . . . . . . . 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 6778	. . . . . . . 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 7278	. . . . . . . 8 ⊢ ((1st ‘𝑤)(1st ‘𝑍)(2nd ‘𝑤)) = ((1st ‘𝑍)‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
111	109, 110	eqtr4di 2796	. . . . . . 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 4812	. . . . . 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 6778	. . . . . . 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 7278	. . . . . . 7 ⊢ ((1st ‘𝑤)(1st ‘𝐸)(2nd ‘𝑤)) = ((1st ‘𝐸)‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
115	113, 114	eqtr4di 2796	. . . . . 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 7293	. . . . 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 6778	. . . . . 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 7278	. . . . . 6 ⊢ ((1st ‘𝑤)𝑀(2nd ‘𝑤)) = (𝑀‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
119	117, 118	eqtr4di 2796	. . . . 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 7293	. . . . . . 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 7870	. . . . . . . 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 6781	. . . . . 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 7278	. . . . . 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 2796	. . . . 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 7296	. . . 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 6778	. . . . . . . 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 7278	. . . . . . . 8 ⊢ ((1st ‘𝑧)(1st ‘𝐸)(2nd ‘𝑧)) = ((1st ‘𝐸)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
129	127, 128	eqtr4di 2796	. . . . . . 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 4812	. . . . . 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 7293	. . . . 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 7293	. . . . . . 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 6781	. . . . . 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 7278	. . . . . 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 2796	. . . . 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 6778	. . . . . 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 7278	. . . . . 6 ⊢ ((1st ‘𝑧)𝑀(2nd ‘𝑧)) = (𝑀‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
138	136, 137	eqtr4di 2796	. . . . 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 7296	. . . 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 2788	. . 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 3127	. 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 2738	. . 3 ⊢ ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
143		eqid 2738	. . 3 ⊢ (comp‘𝑇) = (comp‘𝑇)
144	142, 33, 90, 29, 143, 37, 44	isnat2 17664	. 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 710	1 ⊢ (𝜑 → 𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  ⟨cop 4567   class class class wbr 5074   ↦ cmpt 5157   × cxp 5587  ran crn 5590  Rel wrel 5594   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  1^st c1st 7829  2^nd c2nd 7830  tpos ctpos 8041  Xcixp 8685  Basecbs 16912  Hom chom 16973  compcco 16974  Catccat 17373  Idccid 17374  Hom_f chomf 17375  oppCatcoppc 17420   Func cfunc 17569   ∘_func ccofu 17571   Nat cnat 17657   FuncCat cfuc 17658  SetCatcsetc 17790   ×_c cxpc 17885   1^st_F c1stf 17886   2^nd_F c2ndf 17887   ⟨,⟩_F cprf 17888   eval_F cevlf 17927  Hom_Fchof 17966  Yoncyon 17967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-tpos 8042  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-hom 16986  df-cco 16987  df-cat 17377  df-cid 17378  df-homf 17379  df-comf 17380  df-oppc 17421  df-ssc 17522  df-resc 17523  df-subc 17524  df-func 17573  df-cofu 17575  df-nat 17659  df-fuc 17660  df-setc 17791  df-xpc 17889  df-1stf 17890  df-2ndf 17891  df-prf 17892  df-evlf 17931  df-curf 17932  df-hof 17968  df-yon 17969

This theorem is referenced by:  yonedainv  17999