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Theorem yonedalem4c 18237

Description: Lemma for yoneda 18243. (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 = (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)
yonedalem21.f	⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))
yonedalem21.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
yonedalem4.n	⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1^st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2^nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))))
yonedalem4.p	⊢ (𝜑 → 𝐴 ∈ ((1^st ‘𝐹)‘𝑋))

**Assertion**
Ref	Expression
yonedalem4c	⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ (((1^st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))

Distinct variable groups: 𝑓,𝑔,𝑥,𝑦, 1 𝑢,𝑔,𝐴,𝑦 𝑢,𝑓,𝐶,𝑔,𝑥,𝑦 𝑓,𝐸,𝑔,𝑢,𝑦 𝑓,𝐹,𝑔,𝑢,𝑥,𝑦 𝐵,𝑓,𝑔,𝑢,𝑥,𝑦 𝑓,𝑂,𝑔,𝑢,𝑥,𝑦 𝑆,𝑓,𝑔,𝑢,𝑥,𝑦 𝑄,𝑓,𝑔,𝑢,𝑥 𝑇,𝑓,𝑔,𝑢,𝑦 𝜑,𝑓,𝑔,𝑢,𝑥,𝑦 𝑢,𝑅 𝑓,𝑌,𝑔,𝑢,𝑥,𝑦 𝑓,𝑍,𝑔,𝑢,𝑥,𝑦 𝑓,𝑋,𝑔,𝑢,𝑥,𝑦 Allowed substitution hints: 𝐴(𝑥,𝑓) 𝑄(𝑦) 𝑅(𝑥,𝑦,𝑓,𝑔) 𝑇(𝑥) 𝑈(𝑥,𝑦,𝑢,𝑓,𝑔) 1 (𝑢) 𝐸(𝑥) 𝐻(𝑥,𝑦,𝑢,𝑓,𝑔) 𝑁(𝑥,𝑦,𝑢,𝑓,𝑔) 𝑉(𝑥,𝑦,𝑢,𝑓,𝑔) 𝑊(𝑥,𝑦,𝑢,𝑓,𝑔)

Proof of Theorem yonedalem4c Dummy variables ℎ 𝑘 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		yoneda.y	. . . . 5 ⊢ 𝑌 = (Yon‘𝐶)
2		yoneda.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
3		yoneda.1	. . . . 5 ⊢ 1 = (Id‘𝐶)
4		yoneda.o	. . . . 5 ⊢ 𝑂 = (oppCat‘𝐶)
5		yoneda.s	. . . . 5 ⊢ 𝑆 = (SetCat‘𝑈)
6		yoneda.t	. . . . 5 ⊢ 𝑇 = (SetCat‘𝑉)
7		yoneda.q	. . . . 5 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
8		yoneda.h	. . . . 5 ⊢ 𝐻 = (Hom_F‘𝑄)
9		yoneda.r	. . . . 5 ⊢ 𝑅 = ((𝑄 ×_c 𝑂) FuncCat 𝑇)
10		yoneda.e	. . . . 5 ⊢ 𝐸 = (𝑂 eval_F 𝑆)
11		yoneda.z	. . . . 5 ⊢ 𝑍 = (𝐻 ∘_func ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)))
12		yoneda.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
13		yoneda.w	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
14		yoneda.u	. . . . 5 ⊢ (𝜑 → ran (Hom_f ‘𝐶) ⊆ 𝑈)
15		yoneda.v	. . . . 5 ⊢ (𝜑 → (ran (Hom_f ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
16		yonedalem21.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑂 Func 𝑆))
17		yonedalem21.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
18		yonedalem4.n	. . . . 5 ⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1^st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2^nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))))
19		yonedalem4.p	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ((1^st ‘𝐹)‘𝑋))
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	yonedalem4a 18235	. . . 4 ⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑦)‘𝑔)‘𝐴))))
21		oveq1 7368	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦(Hom ‘𝐶)𝑋) = (𝑧(Hom ‘𝐶)𝑋))
22		oveq2 7369	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑋(2^nd ‘𝐹)𝑦) = (𝑋(2^nd ‘𝐹)𝑧))
23	22	fveq1d 6837	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑋(2^nd ‘𝐹)𝑦)‘𝑔) = ((𝑋(2^nd ‘𝐹)𝑧)‘𝑔))
24	23	fveq1d 6837	. . . . . 6 ⊢ (𝑦 = 𝑧 → (((𝑋(2^nd ‘𝐹)𝑦)‘𝑔)‘𝐴) = (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴))
25	21, 24	mpteq12dv 5173	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑦)‘𝑔)‘𝐴)) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)))
26	25	cbvmptv 5190	. . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑦)‘𝑔)‘𝐴))) = (𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)))
27	20, 26	eqtrdi 2788	. . 3 ⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴))))
28	4, 2	oppcbas 17678	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑂)
29		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
30		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
31		relfunc 17823	. . . . . . . . . . . . . . 15 ⊢ Rel (𝑂 Func 𝑆)
32		1st2ndbr 7989	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1^st ‘𝐹)(𝑂 Func 𝑆)(2^nd ‘𝐹))
33	31, 16, 32	sylancr 588	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1^st ‘𝐹)(𝑂 Func 𝑆)(2^nd ‘𝐹))
34	33	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (1^st ‘𝐹)(𝑂 Func 𝑆)(2^nd ‘𝐹))
35	17	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑋 ∈ 𝐵)
36		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
37	28, 29, 30, 34, 35, 36	funcf2 17829	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑋(2^nd ‘𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1^st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)))
38	37	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (𝑋(2^nd ‘𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1^st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)))
39		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋))
40		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
41	40, 4	oppchom 17675	. . . . . . . . . . . 12 ⊢ (𝑋(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑋)
42	39, 41	eleqtrrdi 2848	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑔 ∈ (𝑋(Hom ‘𝑂)𝑧))
43	38, 42	ffvelcdmd 7032	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2^nd ‘𝐹)𝑧)‘𝑔) ∈ (((1^st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)))
44	15	unssbd 4135	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
45	13, 44	ssexd 5262	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ V)
46	45	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑈 ∈ V)
47	46	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑈 ∈ V)
48		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑆) = (Base‘𝑆)
49	28, 48, 33	funcf1 17827	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶(Base‘𝑆))
50	5, 45	setcbas 18039	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
51	50	feq3d 6648	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((1^st ‘𝐹):𝐵⟶𝑈 ↔ (1^st ‘𝐹):𝐵⟶(Base‘𝑆)))
52	49, 51	mpbird 257	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶𝑈)
53	52, 17	ffvelcdmd 7032	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑋) ∈ 𝑈)
54	53	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1^st ‘𝐹)‘𝑋) ∈ 𝑈)
55	52	ffvelcdmda 7031	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1^st ‘𝐹)‘𝑧) ∈ 𝑈)
56	55	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1^st ‘𝐹)‘𝑧) ∈ 𝑈)
57	5, 47, 30, 54, 56	elsetchom 18042	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔) ∈ (((1^st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)) ↔ ((𝑋(2^nd ‘𝐹)𝑧)‘𝑔):((1^st ‘𝐹)‘𝑋)⟶((1^st ‘𝐹)‘𝑧)))
58	43, 57	mpbid 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2^nd ‘𝐹)𝑧)‘𝑔):((1^st ‘𝐹)‘𝑋)⟶((1^st ‘𝐹)‘𝑧))
59	19	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐴 ∈ ((1^st ‘𝐹)‘𝑋))
60	58, 59	ffvelcdmd 7032	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴) ∈ ((1^st ‘𝐹)‘𝑧))
61	60	fmpttd 7062	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)):(𝑧(Hom ‘𝐶)𝑋)⟶((1^st ‘𝐹)‘𝑧))
62	12	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐶 ∈ Cat)
63	1, 2, 62, 35, 40, 36	yon11 18224	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧) = (𝑧(Hom ‘𝐶)𝑋))
64	63	feq2d 6647	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘𝐹)‘𝑧) ↔ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)):(𝑧(Hom ‘𝐶)𝑋)⟶((1^st ‘𝐹)‘𝑧)))
65	61, 64	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘𝐹)‘𝑧))
66	1, 2, 12, 17, 4, 5, 45, 14	yon1cl 18223	. . . . . . . . . . 11 ⊢ (𝜑 → ((1^st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
67		1st2ndbr 7989	. . . . . . . . . . 11 ⊢ ((Rel (𝑂 Func 𝑆) ∧ ((1^st ‘𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1^st ‘((1^st ‘𝑌)‘𝑋))(𝑂 Func 𝑆)(2^nd ‘((1^st ‘𝑌)‘𝑋)))
68	31, 66, 67	sylancr 588	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘((1^st ‘𝑌)‘𝑋))(𝑂 Func 𝑆)(2^nd ‘((1^st ‘𝑌)‘𝑋)))
69	28, 48, 68	funcf1 17827	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘((1^st ‘𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
70	50	feq3d 6648	. . . . . . . . 9 ⊢ (𝜑 → ((1^st ‘((1^st ‘𝑌)‘𝑋)):𝐵⟶𝑈 ↔ (1^st ‘((1^st ‘𝑌)‘𝑋)):𝐵⟶(Base‘𝑆)))
71	69, 70	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘((1^st ‘𝑌)‘𝑋)):𝐵⟶𝑈)
72	71	ffvelcdmda 7031	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧) ∈ 𝑈)
73	5, 46, 30, 72, 55	elsetchom 18042	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)) ↔ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘𝐹)‘𝑧)))
74	65, 73	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)))
75	74	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)))
76	2	fvexi 6849	. . . . 5 ⊢ 𝐵 ∈ V
77		mptelixpg 8877	. . . . 5 ⊢ (𝐵 ∈ V → ((𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) ∈ X𝑧 ∈ 𝐵 (((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧))))
78	76, 77	ax-mp 5	. . . 4 ⊢ ((𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) ∈ X𝑧 ∈ 𝐵 (((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)))
79	75, 78	sylibr 234	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) ∈ X𝑧 ∈ 𝐵 (((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)))
80	27, 79	eqeltrd 2837	. 2 ⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ X𝑧 ∈ 𝐵 (((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)))
81	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝐶 ∈ Cat)
82	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑋 ∈ 𝐵)
83		simpr1 1196	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑧 ∈ 𝐵)
84	1, 2, 81, 82, 40, 83	yon11 18224	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧) = (𝑧(Hom ‘𝐶)𝑋))
85	84	eleq2d 2823	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑘 ∈ ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧) ↔ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)))
86	85	biimpa 476	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋))
87		eqid 2737	. . . . . . . . . . . 12 ⊢ (comp‘𝑂) = (comp‘𝑂)
88		eqid 2737	. . . . . . . . . . . 12 ⊢ (comp‘𝑆) = (comp‘𝑆)
89	33	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1^st ‘𝐹)(𝑂 Func 𝑆)(2^nd ‘𝐹))
90	89	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (1^st ‘𝐹)(𝑂 Func 𝑆)(2^nd ‘𝐹))
91	82	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑋 ∈ 𝐵)
92	83	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑧 ∈ 𝐵)
93		simpr2 1197	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑤 ∈ 𝐵)
94	93	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑤 ∈ 𝐵)
95		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋))
96	95, 41	eleqtrrdi 2848	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑘 ∈ (𝑋(Hom ‘𝑂)𝑧))
97		simplr3 1219	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))
98	28, 29, 87, 88, 90, 91, 92, 94, 96, 97	funcco 17832	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2^nd ‘𝐹)𝑤)‘(ℎ(⟨𝑋, 𝑧⟩(comp‘𝑂)𝑤)𝑘)) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝑆)((1^st ‘𝐹)‘𝑤))((𝑋(2^nd ‘𝐹)𝑧)‘𝑘)))
99		eqid 2737	. . . . . . . . . . . . 13 ⊢ (comp‘𝐶) = (comp‘𝐶)
100	2, 99, 4, 91, 92, 94	oppcco 17677	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (ℎ(⟨𝑋, 𝑧⟩(comp‘𝑂)𝑤)𝑘) = (𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)ℎ))
101	100	fveq2d 6839	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2^nd ‘𝐹)𝑤)‘(ℎ(⟨𝑋, 𝑧⟩(comp‘𝑂)𝑤)𝑘)) = ((𝑋(2^nd ‘𝐹)𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)ℎ)))
102	45	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑈 ∈ V)
103	102	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑈 ∈ V)
104	53	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1^st ‘𝐹)‘𝑋) ∈ 𝑈)
105	55	3ad2antr1 1190	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1^st ‘𝐹)‘𝑧) ∈ 𝑈)
106	105	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1^st ‘𝐹)‘𝑧) ∈ 𝑈)
107	52	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1^st ‘𝐹):𝐵⟶𝑈)
108	107, 93	ffvelcdmd 7032	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1^st ‘𝐹)‘𝑤) ∈ 𝑈)
109	108	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1^st ‘𝐹)‘𝑤) ∈ 𝑈)
110	28, 29, 30, 89, 82, 83	funcf2 17829	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑋(2^nd ‘𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1^st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)))
111	110	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (𝑋(2^nd ‘𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1^st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)))
112	111, 96	ffvelcdmd 7032	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2^nd ‘𝐹)𝑧)‘𝑘) ∈ (((1^st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)))
113	5, 103, 30, 104, 106	elsetchom 18042	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2^nd ‘𝐹)𝑧)‘𝑘) ∈ (((1^st ‘𝐹)‘𝑋)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)) ↔ ((𝑋(2^nd ‘𝐹)𝑧)‘𝑘):((1^st ‘𝐹)‘𝑋)⟶((1^st ‘𝐹)‘𝑧)))
114	112, 113	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2^nd ‘𝐹)𝑧)‘𝑘):((1^st ‘𝐹)‘𝑋)⟶((1^st ‘𝐹)‘𝑧))
115	28, 29, 30, 89, 83, 93	funcf2 17829	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑧(2^nd ‘𝐹)𝑤):(𝑧(Hom ‘𝑂)𝑤)⟶(((1^st ‘𝐹)‘𝑧)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑤)))
116		simpr3 1198	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))
117	115, 116	ffvelcdmd 7032	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2^nd ‘𝐹)𝑤)‘ℎ) ∈ (((1^st ‘𝐹)‘𝑧)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑤)))
118	5, 102, 30, 105, 108	elsetchom 18042	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ) ∈ (((1^st ‘𝐹)‘𝑧)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑤)) ↔ ((𝑧(2^nd ‘𝐹)𝑤)‘ℎ):((1^st ‘𝐹)‘𝑧)⟶((1^st ‘𝐹)‘𝑤)))
119	117, 118	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2^nd ‘𝐹)𝑤)‘ℎ):((1^st ‘𝐹)‘𝑧)⟶((1^st ‘𝐹)‘𝑤))
120	119	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑧(2^nd ‘𝐹)𝑤)‘ℎ):((1^st ‘𝐹)‘𝑧)⟶((1^st ‘𝐹)‘𝑤))
121	5, 103, 88, 104, 106, 109, 114, 120	setcco 18044	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝑆)((1^st ‘𝐹)‘𝑤))((𝑋(2^nd ‘𝐹)𝑧)‘𝑘)) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ) ∘ ((𝑋(2^nd ‘𝐹)𝑧)‘𝑘)))
122	98, 101, 121	3eqtr3d 2780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2^nd ‘𝐹)𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)ℎ)) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ) ∘ ((𝑋(2^nd ‘𝐹)𝑧)‘𝑘)))
123	122	fveq1d 6837	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2^nd ‘𝐹)𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)ℎ))‘𝐴) = ((((𝑧(2^nd ‘𝐹)𝑤)‘ℎ) ∘ ((𝑋(2^nd ‘𝐹)𝑧)‘𝑘))‘𝐴))
124	19	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐴 ∈ ((1^st ‘𝐹)‘𝑋))
125		fvco3 6934	. . . . . . . . . 10 ⊢ ((((𝑋(2^nd ‘𝐹)𝑧)‘𝑘):((1^st ‘𝐹)‘𝑋)⟶((1^st ‘𝐹)‘𝑧) ∧ 𝐴 ∈ ((1^st ‘𝐹)‘𝑋)) → ((((𝑧(2^nd ‘𝐹)𝑤)‘ℎ) ∘ ((𝑋(2^nd ‘𝐹)𝑧)‘𝑘))‘𝐴) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)‘(((𝑋(2^nd ‘𝐹)𝑧)‘𝑘)‘𝐴)))
126	114, 124, 125	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝑧(2^nd ‘𝐹)𝑤)‘ℎ) ∘ ((𝑋(2^nd ‘𝐹)𝑧)‘𝑘))‘𝐴) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)‘(((𝑋(2^nd ‘𝐹)𝑧)‘𝑘)‘𝐴)))
127	123, 126	eqtrd 2772	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2^nd ‘𝐹)𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)ℎ))‘𝐴) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)‘(((𝑋(2^nd ‘𝐹)𝑧)‘𝑘)‘𝐴)))
128	81	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐶 ∈ Cat)
129	40, 4	oppchom 17675	. . . . . . . . . . . 12 ⊢ (𝑧(Hom ‘𝑂)𝑤) = (𝑤(Hom ‘𝐶)𝑧)
130	97, 129	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ℎ ∈ (𝑤(Hom ‘𝐶)𝑧))
131	1, 2, 128, 91, 40, 92, 99, 94, 130, 95	yon12 18225	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘) = (𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)ℎ))
132	131	fveq2d 6839	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘)) = ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)ℎ)))
133	13	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑉 ∈ 𝑊)
134	14	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ran (Hom_f ‘𝐶) ⊆ 𝑈)
135	15	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (ran (Hom_f ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
136	16	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐹 ∈ (𝑂 Func 𝑆))
137	2, 40, 99, 128, 94, 92, 91, 130, 95	catcocl 17645	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)ℎ) ∈ (𝑤(Hom ‘𝐶)𝑋))
138	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 128, 133, 134, 135, 136, 91, 18, 124, 94, 137	yonedalem4b 18236	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)ℎ)) = (((𝑋(2^nd ‘𝐹)𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)ℎ))‘𝐴))
139	132, 138	eqtrd 2772	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘)) = (((𝑋(2^nd ‘𝐹)𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)ℎ))‘𝐴))
140	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 128, 133, 134, 135, 136, 91, 18, 124, 92, 95	yonedalem4b 18236	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘) = (((𝑋(2^nd ‘𝐹)𝑧)‘𝑘)‘𝐴))
141	140	fveq2d 6839	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘)) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)‘(((𝑋(2^nd ‘𝐹)𝑧)‘𝑘)‘𝐴)))
142	127, 139, 141	3eqtr4d 2782	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘)) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘)))
143	86, 142	syldan 592	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘)) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘)))
144	143	mpteq2dva 5179	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑘 ∈ ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧) ↦ ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘))) = (𝑘 ∈ ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧) ↦ (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘))))
145		fveq2 6835	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = (((𝐹𝑁𝑋)‘𝐴)‘𝑤))
146		fveq2 6835	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧) = ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤))
147		fveq2 6835	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((1^st ‘𝐹)‘𝑧) = ((1^st ‘𝐹)‘𝑤))
148	145, 146, 147	feq123d 6652	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘𝐹)‘𝑧) ↔ (((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤)⟶((1^st ‘𝐹)‘𝑤)))
149	27	fveq1d 6837	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = ((𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)))‘𝑧))
150		ovex 7394	. . . . . . . . . . . . . 14 ⊢ (𝑧(Hom ‘𝐶)𝑋) ∈ V
151	150	mptex 7172	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ V
152		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴))) = (𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)))
153	152	fvmpt2 6954	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐵 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)) ∈ V) → ((𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)))‘𝑧) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)))
154	151, 153	mpan2 692	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐵 → ((𝑧 ∈ 𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)))‘𝑧) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)))
155	149, 154	sylan9eq 2792	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)))
156	155	feq1d 6645	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘𝐹)‘𝑧) ↔ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2^nd ‘𝐹)𝑧)‘𝑔)‘𝐴)):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘𝐹)‘𝑧)))
157	65, 156	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘𝐹)‘𝑧))
158	157	ralrimiva 3130	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘𝐹)‘𝑧))
159	158	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ∀𝑧 ∈ 𝐵 (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘𝐹)‘𝑧))
160	148, 159, 93	rspcdva 3566	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤)⟶((1^st ‘𝐹)‘𝑤))
161	68	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1^st ‘((1^st ‘𝑌)‘𝑋))(𝑂 Func 𝑆)(2^nd ‘((1^st ‘𝑌)‘𝑋)))
162	28, 29, 30, 161, 83, 93	funcf2 17829	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤):(𝑧(Hom ‘𝑂)𝑤)⟶(((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤)))
163	162, 116	ffvelcdmd 7032	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ) ∈ (((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤)))
164	72	3ad2antr1 1190	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧) ∈ 𝑈)
165	71	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1^st ‘((1^st ‘𝑌)‘𝑋)):𝐵⟶𝑈)
166	165, 93	ffvelcdmd 7032	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤) ∈ 𝑈)
167	5, 102, 30, 164, 166	elsetchom 18042	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ) ∈ (((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤)) ↔ ((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤)))
168	163, 167	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤))
169		fcompt 7081	. . . . . 6 ⊢ (((((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤)⟶((1^st ‘𝐹)‘𝑤) ∧ ((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤) ∘ ((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)) = (𝑘 ∈ ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧) ↦ ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘))))
170	160, 168, 169	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤) ∘ ((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)) = (𝑘 ∈ ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧) ↦ ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)‘𝑘))))
171	157	3ad2antr1 1190	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘𝐹)‘𝑧))
172		fcompt 7081	. . . . . 6 ⊢ ((((𝑧(2^nd ‘𝐹)𝑤)‘ℎ):((1^st ‘𝐹)‘𝑧)⟶((1^st ‘𝐹)‘𝑤) ∧ (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)⟶((1^st ‘𝐹)‘𝑧)) → (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ) ∘ (((𝐹𝑁𝑋)‘𝐴)‘𝑧)) = (𝑘 ∈ ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧) ↦ (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘))))
173	119, 171, 172	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ) ∘ (((𝐹𝑁𝑋)‘𝐴)‘𝑧)) = (𝑘 ∈ ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧) ↦ (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘))))
174	144, 170, 173	3eqtr4d 2782	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤) ∘ ((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ) ∘ (((𝐹𝑁𝑋)‘𝐴)‘𝑧)))
175	5, 102, 88, 164, 166, 108, 168, 160	setcco 18044	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)(⟨((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤)⟩(comp‘𝑆)((1^st ‘𝐹)‘𝑤))((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)) = ((((𝐹𝑁𝑋)‘𝐴)‘𝑤) ∘ ((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)))
176	5, 102, 88, 164, 105, 108, 171, 119	setcco 18044	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)(⟨((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝑆)((1^st ‘𝐹)‘𝑤))(((𝐹𝑁𝑋)‘𝐴)‘𝑧)) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ) ∘ (((𝐹𝑁𝑋)‘𝐴)‘𝑧)))
177	174, 175, 176	3eqtr4d 2782	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ∧ ℎ ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)(⟨((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤)⟩(comp‘𝑆)((1^st ‘𝐹)‘𝑤))((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)(⟨((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝑆)((1^st ‘𝐹)‘𝑤))(((𝐹𝑁𝑋)‘𝐴)‘𝑧)))
178	177	ralrimivvva 3184	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∀ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)((((𝐹𝑁𝑋)‘𝐴)‘𝑤)(⟨((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤)⟩(comp‘𝑆)((1^st ‘𝐹)‘𝑤))((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)(⟨((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝑆)((1^st ‘𝐹)‘𝑤))(((𝐹𝑁𝑋)‘𝐴)‘𝑧)))
179		eqid 2737	. . 3 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
180	179, 28, 29, 30, 88, 66, 16	isnat2 17912	. 2 ⊢ (𝜑 → (((𝐹𝑁𝑋)‘𝐴) ∈ (((1^st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↔ (((𝐹𝑁𝑋)‘𝐴) ∈ X𝑧 ∈ 𝐵 (((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1^st ‘𝐹)‘𝑧)) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∀ℎ ∈ (𝑧(Hom ‘𝑂)𝑤)((((𝐹𝑁𝑋)‘𝐴)‘𝑤)(⟨((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧), ((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑤)⟩(comp‘𝑆)((1^st ‘𝐹)‘𝑤))((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑋))𝑤)‘ℎ)) = (((𝑧(2^nd ‘𝐹)𝑤)‘ℎ)(⟨((1^st ‘((1^st ‘𝑌)‘𝑋))‘𝑧), ((1^st ‘𝐹)‘𝑧)⟩(comp‘𝑆)((1^st ‘𝐹)‘𝑤))(((𝐹𝑁𝑋)‘𝐴)‘𝑧)))))
181	80, 178, 180	mpbir2and 714	1 ⊢ (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ (((1^st ‘𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ 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 ran crn 5626 ∘ ccom 5629 Rel wrel 5630 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 ∈ cmpo 7363 1^st c1st 7934 2^nd c2nd 7935 tpos ctpos 8169 Xcixp 8839 Basecbs 17173 Hom chom 17225 compcco 17226 Catccat 17624 Idccid 17625 Hom_f chomf 17626 oppCatcoppc 17671 Func cfunc 17815 ∘_func ccofu 17817 Nat cnat 17905 FuncCat cfuc 17906 SetCatcsetc 18036 ×_c cxpc 18128 1^st_F c1stf 18129 2^nd_F c2ndf 18130 ⟨,⟩_F cprf 18131 eval_F cevlf 18169 Hom_Fchof 18208 Yoncyon 18209

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 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

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 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-fz 13456 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-hom 17238 df-cco 17239 df-cat 17628 df-cid 17629 df-homf 17630 df-comf 17631 df-oppc 17672 df-func 17819 df-nat 17907 df-fuc 17908 df-setc 18037 df-xpc 18132 df-curf 18174 df-hof 18210 df-yon 18211

This theorem is referenced by: yonedainv 18241

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