Theorem yonedalem3 18241

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)
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 7395	. . . . . 6 ⊢ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ∈ V
3	2	mptex 7173	. . . . 5 ⊢ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))) ∈ V
4	1, 3	fnmpoi 8018	. . . 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 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑔 ∈ (𝑂 Func 𝑆))
26		simprr 773	. . . . . . . 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 18235	. . . . . . 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 2737	. . . . . . 7 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
30		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
31	12	fucbas 17925	. . . . . . . . . . 11 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
32	9, 7	oppcbas 17679	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑂)
33	30, 31, 32	xpcbas 18139	. . . . . . . . . 10 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
34		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑇) = (Base‘𝑇)
35		relfunc 17824	. . . . . . . . . . 11 ⊢ Rel ((𝑄 ×c 𝑂) Func 𝑇)
36	6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23	yonedalem1 18233	. . . . . . . . . . . 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 7990	. . . . . . . . . . 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 588	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝑍)((𝑄 ×c 𝑂) Func 𝑇)(2nd ‘𝑍))
40	33, 34, 39	funcf1 17828	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
41	40	fovcdmda 7533	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1st ‘𝑍)𝑦) ∈ (Base‘𝑇))
42	11, 20	setcbas 18040	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑉 = (Base‘𝑇))
43	41, 42	eleqtrrd 2840	. . . . . . 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 7990	. . . . . . . . . . 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 588	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐸)((𝑄 ×c 𝑂) Func 𝑇)(2nd ‘𝐸))
47	33, 34, 46	funcf1 17828	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
48	47	fovcdmda 7533	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1st ‘𝐸)𝑦) ∈ (Base‘𝑇))
49	48, 42	eleqtrrd 2840	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1st ‘𝐸)𝑦) ∈ 𝑉)
50	11, 20, 29, 43, 49	elsetchom 18043	. . . . . 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 3181	. . . 4 ⊢ (𝜑 → ∀𝑔 ∈ (𝑂 Func 𝑆)∀𝑦 ∈ 𝐵 (𝑔𝑀𝑦) ∈ ((𝑔(1st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1st ‘𝐸)𝑦)))
53		fveq2 6836	. . . . . . 7 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀‘𝑧) = (𝑀‘⟨𝑔, 𝑦⟩))
54		df-ov 7365	. . . . . . 7 ⊢ (𝑔𝑀𝑦) = (𝑀‘⟨𝑔, 𝑦⟩)
55	53, 54	eqtr4di 2790	. . . . . 6 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀‘𝑧) = (𝑔𝑀𝑦))
56		fveq2 6836	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st ‘𝑍)‘𝑧) = ((1st ‘𝑍)‘⟨𝑔, 𝑦⟩))
57		df-ov 7365	. . . . . . . 8 ⊢ (𝑔(1st ‘𝑍)𝑦) = ((1st ‘𝑍)‘⟨𝑔, 𝑦⟩)
58	56, 57	eqtr4di 2790	. . . . . . 7 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st ‘𝑍)‘𝑧) = (𝑔(1st ‘𝑍)𝑦))
59		fveq2 6836	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st ‘𝐸)‘𝑧) = ((1st ‘𝐸)‘⟨𝑔, 𝑦⟩))
60		df-ov 7365	. . . . . . . 8 ⊢ (𝑔(1st ‘𝐸)𝑦) = ((1st ‘𝐸)‘⟨𝑔, 𝑦⟩)
61	59, 60	eqtr4di 2790	. . . . . . 7 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1st ‘𝐸)‘𝑧) = (𝑔(1st ‘𝐸)𝑦))
62	58, 61	oveq12d 7380	. . . . . 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 2831	. . . . 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 5792	. . . 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 7395	. . . . . 6 ⊢ (𝑂 Func 𝑆) ∈ V
67	7	fvexi 6850	. . . . . 6 ⊢ 𝐵 ∈ V
68	66, 67	mpoex 8027	. . . . 5 ⊢ (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥)))) ∈ V
69	1, 68	eqeltri 2833	. . . 4 ⊢ 𝑀 ∈ V
70	69	elixp 8847	. . 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 584	. 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 1196	. . . . . 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 7969	. . . . . 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 7970	. . . . . 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 1197	. . . . . 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 7969	. . . . . 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 7970	. . . . . 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 1198	. . . . . . 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 2737	. . . . . . . . . 10 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
88	12, 87	fuchom 17926	. . . . . . . . 9 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
89		eqid 2737	. . . . . . . . 9 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
90		eqid 2737	. . . . . . . . 9 ⊢ (Hom ‘(𝑄 ×c 𝑂)) = (Hom ‘(𝑄 ×c 𝑂))
91	30, 33, 88, 89, 90, 76, 81	xpchom 18141	. . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
93	92, 9	oppchom 17676	. . . . . . . . 9 ⊢ ((2nd ‘𝑧)(Hom ‘𝑂)(2nd ‘𝑤)) = ((2nd ‘𝑤)(Hom ‘𝐶)(2nd ‘𝑧))
94	93	xpeq2i 5653	. . . . . . . 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 2788	. . . . . . 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 2839	. . . . . 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 7969	. . . . . 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 7970	. . . . . 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 18240	. . . 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 7976	. . . . . . . . . 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 6840	. . . . . . . 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 7365	. . . . . . . 8 ⊢ ((1st ‘𝑧)(1st ‘𝑍)(2nd ‘𝑧)) = ((1st ‘𝑍)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
106	104, 105	eqtr4di 2790	. . . . . . 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 7976	. . . . . . . . . 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 6840	. . . . . . . 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 7365	. . . . . . . 8 ⊢ ((1st ‘𝑤)(1st ‘𝑍)(2nd ‘𝑤)) = ((1st ‘𝑍)‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
111	109, 110	eqtr4di 2790	. . . . . . 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 4825	. . . . . 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 6840	. . . . . . 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 7365	. . . . . . 7 ⊢ ((1st ‘𝑤)(1st ‘𝐸)(2nd ‘𝑤)) = ((1st ‘𝐸)‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
115	113, 114	eqtr4di 2790	. . . . . 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 7380	. . . . 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 6840	. . . . . 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 7365	. . . . . 6 ⊢ ((1st ‘𝑤)𝑀(2nd ‘𝑤)) = (𝑀‘⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
119	117, 118	eqtr4di 2790	. . . . 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 7380	. . . . . . 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 7976	. . . . . . . 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 6843	. . . . . 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 7365	. . . . . 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 2790	. . . . 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 7383	. . . 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 6840	. . . . . . . 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 7365	. . . . . . . 8 ⊢ ((1st ‘𝑧)(1st ‘𝐸)(2nd ‘𝑧)) = ((1st ‘𝐸)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
129	127, 128	eqtr4di 2790	. . . . . . 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 4825	. . . . . 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 7380	. . . . 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 7380	. . . . . . 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 6843	. . . . . 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 7365	. . . . . 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 2790	. . . . 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 6840	. . . . . 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 7365	. . . . . 6 ⊢ ((1st ‘𝑧)𝑀(2nd ‘𝑧)) = (𝑀‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
138	136, 137	eqtr4di 2790	. . . . 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 7383	. . . 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 2782	. . 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 3184	. 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 2737	. . 3 ⊢ ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
143		eqid 2737	. . 3 ⊢ (comp‘𝑇) = (comp‘𝑇)
144	142, 33, 90, 29, 143, 37, 44	isnat2 17913	. 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 714	1 ⊢ (𝜑 → 𝑀 ∈ (𝑍((𝑄 ×c 𝑂) Nat 𝑇)𝐸))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ∪ cun 3888   ⊆ wss 3890  ⟨cop 4574   class class class wbr 5086   ↦ cmpt 5167   × cxp 5624  ran crn 5627  Rel wrel 5631   Fn wfn 6489  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362   ∈ cmpo 7364  1^st c1st 7935  2^nd c2nd 7936  tpos ctpos 8170  Xcixp 8840  Basecbs 17174  Hom chom 17226  compcco 17227  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-pm 8771  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-ress 17196  df-hom 17239  df-cco 17240  df-cat 17629  df-cid 17630  df-homf 17631  df-comf 17632  df-oppc 17673  df-ssc 17772  df-resc 17773  df-subc 17774  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-evlf 18174  df-curf 18175  df-hof 18211  df-yon 18212

This theorem is referenced by:  yonedainv  18242