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Theorem yonedalem3 18248

Description: Lemma for yoneda 18251. (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 = (Hom_F‘𝑄)
yoneda.r	⊢ 𝑅 = ((𝑄 ×_c 𝑂) FuncCat 𝑇)
yoneda.e	⊢ 𝐸 = (𝑂 eval_F 𝑆)
yoneda.z	⊢ 𝑍 = (𝐻 ∘_func ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)))
yoneda.c	⊢ (𝜑 → 𝐶 ∈ Cat)
yoneda.w	⊢ (𝜑 → 𝑉 ∈ 𝑊)
yoneda.u	⊢ (𝜑 → ran (Hom_f ‘𝐶) ⊆ 𝑈)
yoneda.v	⊢ (𝜑 → (ran (Hom_f ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
yoneda.m	⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1^st ‘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 ∈ (((1^st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))))
2		ovex 7423	. . . . . 6 ⊢ (((1^st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ∈ V
3	2	mptex 7200	. . . . 5 ⊢ (𝑎 ∈ (((1^st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))) ∈ V
4	1, 3	fnmpoi 8052	. . . 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 = (Hom_F‘𝑄)
14		yoneda.r	. . . . . . . 8 ⊢ 𝑅 = ((𝑄 ×_c 𝑂) FuncCat 𝑇)
15		yoneda.e	. . . . . . . 8 ⊢ 𝐸 = (𝑂 eval_F 𝑆)
16		yoneda.z	. . . . . . . 8 ⊢ 𝑍 = (𝐻 ∘_func ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 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 (Hom_f ‘𝐶) ⊆ 𝑈)
22	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → ran (Hom_f ‘𝐶) ⊆ 𝑈)
23		yoneda.v	. . . . . . . . 9 ⊢ (𝜑 → (ran (Hom_f ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
24	23	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (ran (Hom_f ‘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 18242	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → ((𝑔𝑀𝑦) = (𝑎 ∈ (((1^st ‘𝑌)‘𝑦)(𝑂 Nat 𝑆)𝑔) ↦ ((𝑎‘𝑦)‘( 1 ‘𝑦))) ∧ (𝑔𝑀𝑦):(𝑔(1^st ‘𝑍)𝑦)⟶(𝑔(1^st ‘𝐸)𝑦)))
28	27	simprd 495	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔𝑀𝑦):(𝑔(1^st ‘𝑍)𝑦)⟶(𝑔(1^st ‘𝐸)𝑦))
29		eqid 2730	. . . . . . 7 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
30		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑄 ×_c 𝑂) = (𝑄 ×_c 𝑂)
31	12	fucbas 17932	. . . . . . . . . . 11 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
32	9, 7	oppcbas 17686	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑂)
33	30, 31, 32	xpcbas 18146	. . . . . . . . . 10 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×_c 𝑂))
34		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝑇) = (Base‘𝑇)
35		relfunc 17831	. . . . . . . . . . 11 ⊢ Rel ((𝑄 ×_c 𝑂) Func 𝑇)
36	6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23	yonedalem1 18240	. . . . . . . . . . . 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 8024	. . . . . . . . . . 11 ⊢ ((Rel ((𝑄 ×_c 𝑂) Func 𝑇) ∧ 𝑍 ∈ ((𝑄 ×_c 𝑂) Func 𝑇)) → (1^st ‘𝑍)((𝑄 ×_c 𝑂) Func 𝑇)(2^nd ‘𝑍))
39	35, 37, 38	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘𝑍)((𝑄 ×_c 𝑂) Func 𝑇)(2^nd ‘𝑍))
40	33, 34, 39	funcf1 17835	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝑍):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
41	40	fovcdmda 7563	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1^st ‘𝑍)𝑦) ∈ (Base‘𝑇))
42	11, 20	setcbas 18047	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → 𝑉 = (Base‘𝑇))
43	41, 42	eleqtrrd 2832	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1^st ‘𝑍)𝑦) ∈ 𝑉)
44	36	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×_c 𝑂) Func 𝑇))
45		1st2ndbr 8024	. . . . . . . . . . 11 ⊢ ((Rel ((𝑄 ×_c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×_c 𝑂) Func 𝑇)) → (1^st ‘𝐸)((𝑄 ×_c 𝑂) Func 𝑇)(2^nd ‘𝐸))
46	35, 44, 45	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘𝐸)((𝑄 ×_c 𝑂) Func 𝑇)(2^nd ‘𝐸))
47	33, 34, 46	funcf1 17835	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐸):((𝑂 Func 𝑆) × 𝐵)⟶(Base‘𝑇))
48	47	fovcdmda 7563	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1^st ‘𝐸)𝑦) ∈ (Base‘𝑇))
49	48, 42	eleqtrrd 2832	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔(1^st ‘𝐸)𝑦) ∈ 𝑉)
50	11, 20, 29, 43, 49	elsetchom 18050	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → ((𝑔𝑀𝑦) ∈ ((𝑔(1^st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1^st ‘𝐸)𝑦)) ↔ (𝑔𝑀𝑦):(𝑔(1^st ‘𝑍)𝑦)⟶(𝑔(1^st ‘𝐸)𝑦)))
51	28, 50	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝑂 Func 𝑆) ∧ 𝑦 ∈ 𝐵)) → (𝑔𝑀𝑦) ∈ ((𝑔(1^st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1^st ‘𝐸)𝑦)))
52	51	ralrimivva 3181	. . . 4 ⊢ (𝜑 → ∀𝑔 ∈ (𝑂 Func 𝑆)∀𝑦 ∈ 𝐵 (𝑔𝑀𝑦) ∈ ((𝑔(1^st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1^st ‘𝐸)𝑦)))
53		fveq2 6861	. . . . . . 7 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀‘𝑧) = (𝑀‘⟨𝑔, 𝑦⟩))
54		df-ov 7393	. . . . . . 7 ⊢ (𝑔𝑀𝑦) = (𝑀‘⟨𝑔, 𝑦⟩)
55	53, 54	eqtr4di 2783	. . . . . 6 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → (𝑀‘𝑧) = (𝑔𝑀𝑦))
56		fveq2 6861	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1^st ‘𝑍)‘𝑧) = ((1^st ‘𝑍)‘⟨𝑔, 𝑦⟩))
57		df-ov 7393	. . . . . . . 8 ⊢ (𝑔(1^st ‘𝑍)𝑦) = ((1^st ‘𝑍)‘⟨𝑔, 𝑦⟩)
58	56, 57	eqtr4di 2783	. . . . . . 7 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1^st ‘𝑍)‘𝑧) = (𝑔(1^st ‘𝑍)𝑦))
59		fveq2 6861	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1^st ‘𝐸)‘𝑧) = ((1^st ‘𝐸)‘⟨𝑔, 𝑦⟩))
60		df-ov 7393	. . . . . . . 8 ⊢ (𝑔(1^st ‘𝐸)𝑦) = ((1^st ‘𝐸)‘⟨𝑔, 𝑦⟩)
61	59, 60	eqtr4di 2783	. . . . . . 7 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((1^st ‘𝐸)‘𝑧) = (𝑔(1^st ‘𝐸)𝑦))
62	58, 61	oveq12d 7408	. . . . . 6 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → (((1^st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1^st ‘𝐸)‘𝑧)) = ((𝑔(1^st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1^st ‘𝐸)𝑦)))
63	55, 62	eleq12d 2823	. . . . 5 ⊢ (𝑧 = ⟨𝑔, 𝑦⟩ → ((𝑀‘𝑧) ∈ (((1^st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1^st ‘𝐸)‘𝑧)) ↔ (𝑔𝑀𝑦) ∈ ((𝑔(1^st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1^st ‘𝐸)𝑦))))
64	63	ralxp 5808	. . . 4 ⊢ (∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀‘𝑧) ∈ (((1^st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1^st ‘𝐸)‘𝑧)) ↔ ∀𝑔 ∈ (𝑂 Func 𝑆)∀𝑦 ∈ 𝐵 (𝑔𝑀𝑦) ∈ ((𝑔(1^st ‘𝑍)𝑦)(Hom ‘𝑇)(𝑔(1^st ‘𝐸)𝑦)))
65	52, 64	sylibr 234	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀‘𝑧) ∈ (((1^st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1^st ‘𝐸)‘𝑧)))
66		ovex 7423	. . . . . 6 ⊢ (𝑂 Func 𝑆) ∈ V
67	7	fvexi 6875	. . . . . 6 ⊢ 𝐵 ∈ V
68	66, 67	mpoex 8061	. . . . 5 ⊢ (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1^st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥)))) ∈ V
69	1, 68	eqeltri 2825	. . . 4 ⊢ 𝑀 ∈ V
70	69	elixp 8880	. . 3 ⊢ (𝑀 ∈ X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1^st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1^st ‘𝐸)‘𝑧)) ↔ (𝑀 Fn ((𝑂 Func 𝑆) × 𝐵) ∧ ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑀‘𝑧) ∈ (((1^st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1^st ‘𝐸)‘𝑧))))
71	5, 65, 70	sylanbrc 583	. 2 ⊢ (𝜑 → 𝑀 ∈ X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1^st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1^st ‘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 (Hom_f ‘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 (Hom_f ‘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 8003	. . . . . 6 ⊢ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → (1^st ‘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))) → (1^st ‘𝑧) ∈ (𝑂 Func 𝑆))
79		xp2nd 8004	. . . . . 6 ⊢ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → (2^nd ‘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))) → (2^nd ‘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 8003	. . . . . 6 ⊢ (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → (1^st ‘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))) → (1^st ‘𝑤) ∈ (𝑂 Func 𝑆))
84		xp2nd 8004	. . . . . 6 ⊢ (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → (2^nd ‘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))) → (2^nd ‘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 2730	. . . . . . . . . 10 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
88	12, 87	fuchom 17933	. . . . . . . . 9 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
89		eqid 2730	. . . . . . . . 9 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
90		eqid 2730	. . . . . . . . 9 ⊢ (Hom ‘(𝑄 ×_c 𝑂)) = (Hom ‘(𝑄 ×_c 𝑂))
91	30, 33, 88, 89, 90, 76, 81	xpchom 18148	. . . . . . . 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) = (((1^st ‘𝑧)(𝑂 Nat 𝑆)(1^st ‘𝑤)) × ((2^nd ‘𝑧)(Hom ‘𝑂)(2^nd ‘𝑤))))
92		eqid 2730	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
93	92, 9	oppchom 17683	. . . . . . . . 9 ⊢ ((2^nd ‘𝑧)(Hom ‘𝑂)(2^nd ‘𝑤)) = ((2^nd ‘𝑤)(Hom ‘𝐶)(2^nd ‘𝑧))
94	93	xpeq2i 5668	. . . . . . . 8 ⊢ (((1^st ‘𝑧)(𝑂 Nat 𝑆)(1^st ‘𝑤)) × ((2^nd ‘𝑧)(Hom ‘𝑂)(2^nd ‘𝑤))) = (((1^st ‘𝑧)(𝑂 Nat 𝑆)(1^st ‘𝑤)) × ((2^nd ‘𝑤)(Hom ‘𝐶)(2^nd ‘𝑧)))
95	91, 94	eqtrdi 2781	. . . . . . 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) = (((1^st ‘𝑧)(𝑂 Nat 𝑆)(1^st ‘𝑤)) × ((2^nd ‘𝑤)(Hom ‘𝐶)(2^nd ‘𝑧))))
96	86, 95	eleqtrd 2831	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → 𝑔 ∈ (((1^st ‘𝑧)(𝑂 Nat 𝑆)(1^st ‘𝑤)) × ((2^nd ‘𝑤)(Hom ‘𝐶)(2^nd ‘𝑧))))
97		xp1st 8003	. . . . . 6 ⊢ (𝑔 ∈ (((1^st ‘𝑧)(𝑂 Nat 𝑆)(1^st ‘𝑤)) × ((2^nd ‘𝑤)(Hom ‘𝐶)(2^nd ‘𝑧))) → (1^st ‘𝑔) ∈ ((1^st ‘𝑧)(𝑂 Nat 𝑆)(1^st ‘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))) → (1^st ‘𝑔) ∈ ((1^st ‘𝑧)(𝑂 Nat 𝑆)(1^st ‘𝑤)))
99		xp2nd 8004	. . . . . 6 ⊢ (𝑔 ∈ (((1^st ‘𝑧)(𝑂 Nat 𝑆)(1^st ‘𝑤)) × ((2^nd ‘𝑤)(Hom ‘𝐶)(2^nd ‘𝑧))) → (2^nd ‘𝑔) ∈ ((2^nd ‘𝑤)(Hom ‘𝐶)(2^nd ‘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))) → (2^nd ‘𝑔) ∈ ((2^nd ‘𝑤)(Hom ‘𝐶)(2^nd ‘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 18247	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → (((1^st ‘𝑤)𝑀(2^nd ‘𝑤))(⟨((1^st ‘𝑧)(1^st ‘𝑍)(2^nd ‘𝑧)), ((1^st ‘𝑤)(1^st ‘𝑍)(2^nd ‘𝑤))⟩(comp‘𝑇)((1^st ‘𝑤)(1^st ‘𝐸)(2^nd ‘𝑤)))((1^st ‘𝑔)(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝑍)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)(2^nd ‘𝑔))) = (((1^st ‘𝑔)(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝐸)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)(2^nd ‘𝑔))(⟨((1^st ‘𝑧)(1^st ‘𝑍)(2^nd ‘𝑧)), ((1^st ‘𝑧)(1^st ‘𝐸)(2^nd ‘𝑧))⟩(comp‘𝑇)((1^st ‘𝑤)(1^st ‘𝐸)(2^nd ‘𝑤)))((1^st ‘𝑧)𝑀(2^nd ‘𝑧))))
102		1st2nd2 8010	. . . . . . . . . 10 ⊢ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘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 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
104	103	fveq2d 6865	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((1^st ‘𝑍)‘𝑧) = ((1^st ‘𝑍)‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
105		df-ov 7393	. . . . . . . 8 ⊢ ((1^st ‘𝑧)(1^st ‘𝑍)(2^nd ‘𝑧)) = ((1^st ‘𝑍)‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
106	104, 105	eqtr4di 2783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((1^st ‘𝑍)‘𝑧) = ((1^st ‘𝑧)(1^st ‘𝑍)(2^nd ‘𝑧)))
107		1st2nd2 8010	. . . . . . . . . 10 ⊢ (𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘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 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
109	108	fveq2d 6865	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((1^st ‘𝑍)‘𝑤) = ((1^st ‘𝑍)‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
110		df-ov 7393	. . . . . . . 8 ⊢ ((1^st ‘𝑤)(1^st ‘𝑍)(2^nd ‘𝑤)) = ((1^st ‘𝑍)‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
111	109, 110	eqtr4di 2783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((1^st ‘𝑍)‘𝑤) = ((1^st ‘𝑤)(1^st ‘𝑍)(2^nd ‘𝑤)))
112	106, 111	opeq12d 4848	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝑍)‘𝑤)⟩ = ⟨((1^st ‘𝑧)(1^st ‘𝑍)(2^nd ‘𝑧)), ((1^st ‘𝑤)(1^st ‘𝑍)(2^nd ‘𝑤))⟩)
113	108	fveq2d 6865	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((1^st ‘𝐸)‘𝑤) = ((1^st ‘𝐸)‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
114		df-ov 7393	. . . . . . 7 ⊢ ((1^st ‘𝑤)(1^st ‘𝐸)(2^nd ‘𝑤)) = ((1^st ‘𝐸)‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
115	113, 114	eqtr4di 2783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((1^st ‘𝐸)‘𝑤) = ((1^st ‘𝑤)(1^st ‘𝐸)(2^nd ‘𝑤)))
116	112, 115	oveq12d 7408	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → (⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝑍)‘𝑤)⟩(comp‘𝑇)((1^st ‘𝐸)‘𝑤)) = (⟨((1^st ‘𝑧)(1^st ‘𝑍)(2^nd ‘𝑧)), ((1^st ‘𝑤)(1^st ‘𝑍)(2^nd ‘𝑤))⟩(comp‘𝑇)((1^st ‘𝑤)(1^st ‘𝐸)(2^nd ‘𝑤))))
117	108	fveq2d 6865	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → (𝑀‘𝑤) = (𝑀‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
118		df-ov 7393	. . . . . 6 ⊢ ((1^st ‘𝑤)𝑀(2^nd ‘𝑤)) = (𝑀‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
119	117, 118	eqtr4di 2783	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → (𝑀‘𝑤) = ((1^st ‘𝑤)𝑀(2^nd ‘𝑤)))
120	103, 108	oveq12d 7408	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → (𝑧(2^nd ‘𝑍)𝑤) = (⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝑍)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
121		1st2nd2 8010	. . . . . . . 8 ⊢ (𝑔 ∈ (((1^st ‘𝑧)(𝑂 Nat 𝑆)(1^st ‘𝑤)) × ((2^nd ‘𝑤)(Hom ‘𝐶)(2^nd ‘𝑧))) → 𝑔 = ⟨(1^st ‘𝑔), (2^nd ‘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 = ⟨(1^st ‘𝑔), (2^nd ‘𝑔)⟩)
123	120, 122	fveq12d 6868	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((𝑧(2^nd ‘𝑍)𝑤)‘𝑔) = ((⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝑍)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)‘⟨(1^st ‘𝑔), (2^nd ‘𝑔)⟩))
124		df-ov 7393	. . . . . 6 ⊢ ((1^st ‘𝑔)(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝑍)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)(2^nd ‘𝑔)) = ((⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝑍)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)‘⟨(1^st ‘𝑔), (2^nd ‘𝑔)⟩)
125	123, 124	eqtr4di 2783	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((𝑧(2^nd ‘𝑍)𝑤)‘𝑔) = ((1^st ‘𝑔)(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝑍)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)(2^nd ‘𝑔)))
126	116, 119, 125	oveq123d 7411	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((𝑀‘𝑤)(⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝑍)‘𝑤)⟩(comp‘𝑇)((1^st ‘𝐸)‘𝑤))((𝑧(2^nd ‘𝑍)𝑤)‘𝑔)) = (((1^st ‘𝑤)𝑀(2^nd ‘𝑤))(⟨((1^st ‘𝑧)(1^st ‘𝑍)(2^nd ‘𝑧)), ((1^st ‘𝑤)(1^st ‘𝑍)(2^nd ‘𝑤))⟩(comp‘𝑇)((1^st ‘𝑤)(1^st ‘𝐸)(2^nd ‘𝑤)))((1^st ‘𝑔)(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝑍)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)(2^nd ‘𝑔))))
127	103	fveq2d 6865	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((1^st ‘𝐸)‘𝑧) = ((1^st ‘𝐸)‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
128		df-ov 7393	. . . . . . . 8 ⊢ ((1^st ‘𝑧)(1^st ‘𝐸)(2^nd ‘𝑧)) = ((1^st ‘𝐸)‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
129	127, 128	eqtr4di 2783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((1^st ‘𝐸)‘𝑧) = ((1^st ‘𝑧)(1^st ‘𝐸)(2^nd ‘𝑧)))
130	106, 129	opeq12d 4848	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝐸)‘𝑧)⟩ = ⟨((1^st ‘𝑧)(1^st ‘𝑍)(2^nd ‘𝑧)), ((1^st ‘𝑧)(1^st ‘𝐸)(2^nd ‘𝑧))⟩)
131	130, 115	oveq12d 7408	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → (⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝐸)‘𝑧)⟩(comp‘𝑇)((1^st ‘𝐸)‘𝑤)) = (⟨((1^st ‘𝑧)(1^st ‘𝑍)(2^nd ‘𝑧)), ((1^st ‘𝑧)(1^st ‘𝐸)(2^nd ‘𝑧))⟩(comp‘𝑇)((1^st ‘𝑤)(1^st ‘𝐸)(2^nd ‘𝑤))))
132	103, 108	oveq12d 7408	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → (𝑧(2^nd ‘𝐸)𝑤) = (⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝐸)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
133	132, 122	fveq12d 6868	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((𝑧(2^nd ‘𝐸)𝑤)‘𝑔) = ((⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝐸)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)‘⟨(1^st ‘𝑔), (2^nd ‘𝑔)⟩))
134		df-ov 7393	. . . . . 6 ⊢ ((1^st ‘𝑔)(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝐸)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)(2^nd ‘𝑔)) = ((⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝐸)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)‘⟨(1^st ‘𝑔), (2^nd ‘𝑔)⟩)
135	133, 134	eqtr4di 2783	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((𝑧(2^nd ‘𝐸)𝑤)‘𝑔) = ((1^st ‘𝑔)(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝐸)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)(2^nd ‘𝑔)))
136	103	fveq2d 6865	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → (𝑀‘𝑧) = (𝑀‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
137		df-ov 7393	. . . . . 6 ⊢ ((1^st ‘𝑧)𝑀(2^nd ‘𝑧)) = (𝑀‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
138	136, 137	eqtr4di 2783	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → (𝑀‘𝑧) = ((1^st ‘𝑧)𝑀(2^nd ‘𝑧)))
139	131, 135, 138	oveq123d 7411	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → (((𝑧(2^nd ‘𝐸)𝑤)‘𝑔)(⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝐸)‘𝑧)⟩(comp‘𝑇)((1^st ‘𝐸)‘𝑤))(𝑀‘𝑧)) = (((1^st ‘𝑔)(⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩(2^nd ‘𝐸)⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)(2^nd ‘𝑔))(⟨((1^st ‘𝑧)(1^st ‘𝑍)(2^nd ‘𝑧)), ((1^st ‘𝑧)(1^st ‘𝐸)(2^nd ‘𝑧))⟩(comp‘𝑇)((1^st ‘𝑤)(1^st ‘𝐸)(2^nd ‘𝑤)))((1^st ‘𝑧)𝑀(2^nd ‘𝑧))))
140	101, 126, 139	3eqtr4d 2775	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤))) → ((𝑀‘𝑤)(⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝑍)‘𝑤)⟩(comp‘𝑇)((1^st ‘𝐸)‘𝑤))((𝑧(2^nd ‘𝑍)𝑤)‘𝑔)) = (((𝑧(2^nd ‘𝐸)𝑤)‘𝑔)(⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝐸)‘𝑧)⟩(comp‘𝑇)((1^st ‘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)(⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝑍)‘𝑤)⟩(comp‘𝑇)((1^st ‘𝐸)‘𝑤))((𝑧(2^nd ‘𝑍)𝑤)‘𝑔)) = (((𝑧(2^nd ‘𝐸)𝑤)‘𝑔)(⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝐸)‘𝑧)⟩(comp‘𝑇)((1^st ‘𝐸)‘𝑤))(𝑀‘𝑧)))
142		eqid 2730	. . 3 ⊢ ((𝑄 ×_c 𝑂) Nat 𝑇) = ((𝑄 ×_c 𝑂) Nat 𝑇)
143		eqid 2730	. . 3 ⊢ (comp‘𝑇) = (comp‘𝑇)
144	142, 33, 90, 29, 143, 37, 44	isnat2 17920	. 2 ⊢ (𝜑 → (𝑀 ∈ (𝑍((𝑄 ×_c 𝑂) Nat 𝑇)𝐸) ↔ (𝑀 ∈ X𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)(((1^st ‘𝑍)‘𝑧)(Hom ‘𝑇)((1^st ‘𝐸)‘𝑧)) ∧ ∀𝑧 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑤 ∈ ((𝑂 Func 𝑆) × 𝐵)∀𝑔 ∈ (𝑧(Hom ‘(𝑄 ×_c 𝑂))𝑤)((𝑀‘𝑤)(⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝑍)‘𝑤)⟩(comp‘𝑇)((1^st ‘𝐸)‘𝑤))((𝑧(2^nd ‘𝑍)𝑤)‘𝑔)) = (((𝑧(2^nd ‘𝐸)𝑤)‘𝑔)(⟨((1^st ‘𝑍)‘𝑧), ((1^st ‘𝐸)‘𝑧)⟩(comp‘𝑇)((1^st ‘𝐸)‘𝑤))(𝑀‘𝑧)))))
145	71, 141, 144	mpbir2and 713	1 ⊢ (𝜑 → 𝑀 ∈ (𝑍((𝑄 ×_c 𝑂) Nat 𝑇)𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 Vcvv 3450 ∪ cun 3915 ⊆ wss 3917 ⟨cop 4598 class class class wbr 5110 ↦ cmpt 5191 × cxp 5639 ran crn 5642 Rel wrel 5646 Fn wfn 6509 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 1^st c1st 7969 2^nd c2nd 7970 tpos ctpos 8207 Xcixp 8873 Basecbs 17186 Hom chom 17238 compcco 17239 Catccat 17632 Idccid 17633 Hom_f chomf 17634 oppCatcoppc 17679 Func cfunc 17823 ∘_func ccofu 17825 Nat cnat 17913 FuncCat cfuc 17914 SetCatcsetc 18044 ×_c cxpc 18136 1^st_F c1stf 18137 2^nd_F c2ndf 18138 ⟨,⟩_F cprf 18139 eval_F cevlf 18177 Hom_Fchof 18216 Yoncyon 18217

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-hom 17251 df-cco 17252 df-cat 17636 df-cid 17637 df-homf 17638 df-comf 17639 df-oppc 17680 df-ssc 17779 df-resc 17780 df-subc 17781 df-func 17827 df-cofu 17829 df-nat 17915 df-fuc 17916 df-setc 18045 df-xpc 18140 df-1stf 18141 df-2ndf 18142 df-prf 18143 df-evlf 18181 df-curf 18182 df-hof 18218 df-yon 18219

This theorem is referenced by: yonedainv 18249

W3C validator