Theorem yonffthlem 18328

Description: Lemma for yonffth 18330. (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 = (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))))
yonedainv.i	⊢ 𝐼 = (Inv‘𝑅)
yonedainv.n	⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))))

Assertion

Ref	Expression
yonffthlem	⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Distinct variable groups:   𝑓,𝑎,𝑔,𝑥,𝑦, 1   𝑢,𝑎,𝑔,𝑦,𝐶,𝑓,𝑥   𝐸,𝑎,𝑓,𝑔,𝑢,𝑦   𝐵,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑁,𝑎   𝑂,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑆,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑔,𝑀,𝑢,𝑦   𝑄,𝑎,𝑓,𝑔,𝑢,𝑥   𝑇,𝑓,𝑔,𝑢,𝑦   𝜑,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑢,𝑅   𝑌,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦   𝑍,𝑎,𝑓,𝑔,𝑢,𝑥,𝑦

Allowed substitution hints:   𝑄(𝑦)   𝑅(𝑥,𝑦,𝑓,𝑔,𝑎)   𝑇(𝑥,𝑎)   𝑈(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   1 (𝑢)   𝐸(𝑥)   𝐻(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝐼(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝑀(𝑥,𝑓,𝑎)   𝑁(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)   𝑊(𝑥,𝑦,𝑢,𝑓,𝑔,𝑎)

Proof of Theorem yonffthlem

Dummy variables ℎ 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relfunc 17909	. . 3 ⊢ Rel (𝐶 Func 𝑄)
2		yoneda.y	. . . 4 ⊢ 𝑌 = (Yon‘𝐶)
3		yoneda.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		yoneda.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐶)
5		yoneda.s	. . . 4 ⊢ 𝑆 = (SetCat‘𝑈)
6		yoneda.q	. . . 4 ⊢ 𝑄 = (𝑂 FuncCat 𝑆)
7		yoneda.w	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
8		yoneda.v	. . . . . 6 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
9	8	unssbd 4149	. . . . 5 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
10	7, 9	ssexd 5285	. . . 4 ⊢ (𝜑 → 𝑈 ∈ V)
11		yoneda.u	. . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
12	2, 3, 4, 5, 6, 10, 11	yoncl 18308	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
13		1st2nd 8024	. . 3 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
14	1, 12, 13	sylancr 598	. 2 ⊢ (𝜑 → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
15		1st2ndbr 8027	. . . . 5 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
16	1, 12, 15	sylancr 598	. . . 4 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
17		fveq2 6871	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → (𝑁‘𝑣) = (𝑁‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩))
18		df-ov 7403	. . . . . . . . . . 11 ⊢ (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑁‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩)
19	17, 18	eqtr4di 2818	. . . . . . . . . 10 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → (𝑁‘𝑣) = (((1st ‘𝑌)‘𝑤)𝑁𝑧))
20		fveq2 6871	. . . . . . . . . . . 12 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝐸)‘𝑣) = ((1st ‘𝐸)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩))
21		df-ov 7403	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = ((1st ‘𝐸)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩)
22	20, 21	eqtr4di 2818	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝐸)‘𝑣) = (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧))
23		fveq2 6871	. . . . . . . . . . . 12 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝑍)‘𝑣) = ((1st ‘𝑍)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩))
24		df-ov 7403	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧) = ((1st ‘𝑍)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩)
25	23, 24	eqtr4di 2818	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝑍)‘𝑣) = (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧))
26	22, 25	oveq12d 7418	. . . . . . . . . 10 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)) = ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧)))
27	19, 26	eleq12d 2859	. . . . . . . . 9 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)) ↔ (((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧))))
28		yoneda.r	. . . . . . . . . . . . . 14 ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
29	28	fucbas 18010	. . . . . . . . . . . . 13 ⊢ ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅)
30		yonedainv.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (Inv‘𝑅)
31		yoneda.b	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = (Base‘𝐶)
32		yoneda.1	. . . . . . . . . . . . . . . . . 18 ⊢ 1 = (Id‘𝐶)
33		yoneda.t	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑇 = (SetCat‘𝑉)
34		yoneda.h	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐻 = (HomF‘𝑄)
35		yoneda.e	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐸 = (𝑂 evalF 𝑆)
36		yoneda.z	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
37	2, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 3, 7, 11, 8	yonedalem1 18318	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
38	37	simpld 499	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
39		funcrcl 17910	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
40	38, 39	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
41	40	simpld 499	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄 ×c 𝑂) ∈ Cat)
42	40	simprd 500	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ∈ Cat)
43	28, 41, 42	fuccat 18020	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Cat)
44	37	simprd 500	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
45		eqid 2765	. . . . . . . . . . . . 13 ⊢ (Iso‘𝑅) = (Iso‘𝑅)
46		yoneda.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))))
47		yonedainv.n	. . . . . . . . . . . . . 14 ⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))))
48	2, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 3, 7, 11, 8, 46, 30, 47	yonedainv 18327	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀(𝑍𝐼𝐸)𝑁)
49	29, 30, 43, 38, 44, 45, 48	inviso2 17814	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (𝐸(Iso‘𝑅)𝑍))
50		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
51	6	fucbas 18010	. . . . . . . . . . . . . 14 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
52	4, 31	oppcbas 17764	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝑂)
53	50, 51, 52	xpcbas 18224	. . . . . . . . . . . . 13 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
54		eqid 2765	. . . . . . . . . . . . 13 ⊢ ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
55		eqid 2765	. . . . . . . . . . . . 13 ⊢ (Iso‘𝑇) = (Iso‘𝑇)
56	28, 53, 54, 44, 38, 45, 55	fuciso 18025	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 ∈ (𝐸(Iso‘𝑅)𝑍) ↔ (𝑁 ∈ (𝐸((𝑄 ×c 𝑂) Nat 𝑇)𝑍) ∧ ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)))))
57	49, 56	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∈ (𝐸((𝑄 ×c 𝑂) Nat 𝑇)𝑍) ∧ ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣))))
58	57	simprd 500	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)))
59	58	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)))
60	31, 51, 16	funcf1 17913	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆))
61	60	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆))
62		simprr 784	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
63	61, 62	ffvelcdmd 7070	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
64		simprl 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
65	63, 64	opelxpd 5691	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
66	27, 59, 65	rspcdva 3585	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧)))
67	4	oppccat 17768	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
68	3, 67	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂 ∈ Cat)
69	68	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑂 ∈ Cat)
70	5	setccat 18132	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
71	10, 70	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Cat)
72	71	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑆 ∈ Cat)
73	35, 69, 72, 52, 63, 64	evlf1 18266	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧))
74	3	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝐶 ∈ Cat)
75		eqid 2765	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
76	2, 31, 74, 62, 75, 64	yon11 18310	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) = (𝑧(Hom ‘𝐶)𝑤))
77	73, 76	eqtrd 2800	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = (𝑧(Hom ‘𝐶)𝑤))
78	7	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑉 ∈ 𝑊)
79	11	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ran (Homf ‘𝐶) ⊆ 𝑈)
80	8	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
81	2, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 74, 78, 79, 80, 63, 64	yonedalem21 18319	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧) = (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))
82	77, 81	oveq12d 7418	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧)) = ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))))
83	66, 82	eleqtrd 2867	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))))
84	9	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 ⊆ 𝑉)
85		eqid 2765	. . . . . . . . . . . . 13 ⊢ (Base‘𝑆) = (Base‘𝑆)
86		relfunc 17909	. . . . . . . . . . . . . 14 ⊢ Rel (𝑂 Func 𝑆)
87		1st2ndbr 8027	. . . . . . . . . . . . . 14 ⊢ ((Rel (𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑤)))
88	86, 63, 87	sylancr 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘((1st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑤)))
89	52, 85, 88	funcf1 17913	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘((1st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
90	89, 64	ffvelcdmd 7070	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ (Base‘𝑆))
91	5, 10	setcbas 18125	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
92	91	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 = (Base‘𝑆))
93	90, 92	eleqtrrd 2868	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ 𝑈)
94	76, 93	eqeltrrd 2866	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑈)
95	84, 94	sseldd 3940	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑉)
96		eqid 2765	. . . . . . . . . 10 ⊢ (Homf ‘𝑄) = (Homf ‘𝑄)
97		eqid 2765	. . . . . . . . . . 11 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
98	6, 97	fuchom 18011	. . . . . . . . . 10 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
99	61, 64	ffvelcdmd 7070	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
100	96, 51, 98, 99, 63	homfval 17738	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) = (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))
101	8	unssad 4148	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉)
102	101	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ran (Homf ‘𝑄) ⊆ 𝑉)
103	96, 51	homffn 17739	. . . . . . . . . . 11 ⊢ (Homf ‘𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆))
104		fnovrn 7575	. . . . . . . . . . 11 ⊢ (((Homf ‘𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆)) ∧ ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ ran (Homf ‘𝑄))
105	103, 99, 63, 104	mp3an2i 1490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ ran (Homf ‘𝑄))
106	102, 105	sseldd 3940	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ 𝑉)
107	100, 106	eqeltrrd 2866	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)) ∈ 𝑉)
108	33, 78, 95, 107, 55	setciso 18138	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))) ↔ (((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))))
109	83, 108	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))
110	74	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝐶 ∈ Cat)
111	110	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
112	64	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑧 ∈ 𝐵)
113	112	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑧 ∈ 𝐵)
114		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
115	2, 31, 111, 113, 75, 114	yon11 18310	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦) = (𝑦(Hom ‘𝐶)𝑧))
116	115	eqcomd 2771	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (𝑦(Hom ‘𝐶)𝑧) = ((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦))
117	111	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝐶 ∈ Cat)
118	62	ad3antrrr 742	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑤 ∈ 𝐵)
119	113	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑧 ∈ 𝐵)
120		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (comp‘𝐶) = (comp‘𝐶)
121	114	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑦 ∈ 𝐵)
122		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
123		simpllr 787	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ℎ ∈ (𝑧(Hom ‘𝐶)𝑤))
124	2, 31, 117, 118, 75, 119, 120, 121, 122, 123	yon12 18311	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑧(2nd ‘((1st ‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ) = (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔))
125	2, 31, 117, 119, 75, 118, 120, 121, 123, 122	yon2 18312	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔) = (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔))
126	124, 125	eqtr4d 2803	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑧(2nd ‘((1st ‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ) = ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔))
127	116, 126	mpteq12dva 5191	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st ‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ)) = (𝑔 ∈ ((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦) ↦ ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔)))
128	16	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
129	31, 75, 98, 128, 64, 62	funcf2 17915	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))
130	129	ffvelcdmda 7069	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) ∈ (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))
131	97, 130	nat1st2nd 18001	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) ∈ (⟨(1st ‘((1st ‘𝑌)‘𝑧)), (2nd ‘((1st ‘𝑌)‘𝑧))⟩(𝑂 Nat 𝑆)⟨(1st ‘((1st ‘𝑌)‘𝑤)), (2nd ‘((1st ‘𝑌)‘𝑤))⟩))
132	131	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) ∈ (⟨(1st ‘((1st ‘𝑌)‘𝑧)), (2nd ‘((1st ‘𝑌)‘𝑧))⟩(𝑂 Nat 𝑆)⟨(1st ‘((1st ‘𝑌)‘𝑤)), (2nd ‘((1st ‘𝑌)‘𝑤))⟩))
133		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
134	97, 132, 52, 133, 114	natcl 18003	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦) ∈ (((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦)(Hom ‘𝑆)((1st ‘((1st ‘𝑌)‘𝑤))‘𝑦)))
135	10	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 ∈ V)
136	135	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 ∈ V)
137	60	ad2antrr 738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆))
138	137, 112	ffvelcdmd 7070	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
139		1st2ndbr 8027	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Rel (𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st ‘𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑧)))
140	86, 138, 139	sylancr 598	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st ‘𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑧)))
141	52, 85, 140	funcf1 17913	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st ‘𝑌)‘𝑧)):𝐵⟶(Base‘𝑆))
142	141	ffvelcdmda 7069	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦) ∈ (Base‘𝑆))
143	92	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 = (Base‘𝑆))
144	142, 143	eleqtrrd 2868	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦) ∈ 𝑈)
145	89	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
146	145	ffvelcdmda 7069	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑦) ∈ (Base‘𝑆))
147	146, 143	eleqtrrd 2868	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑦) ∈ 𝑈)
148	5, 136, 133, 144, 147	elsetchom 18128	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦) ∈ (((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦)(Hom ‘𝑆)((1st ‘((1st ‘𝑌)‘𝑤))‘𝑦)) ↔ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦):((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦)⟶((1st ‘((1st ‘𝑌)‘𝑤))‘𝑦)))
149	134, 148	mpbid 235	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦):((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦)⟶((1st ‘((1st ‘𝑌)‘𝑤))‘𝑦))
150	149	feqmptd 6939	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦) = (𝑔 ∈ ((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦) ↦ ((((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔)))
151	127, 150	eqtr4d 2803	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st ‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ)) = (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦))
152	151	mpteq2dva 5198	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st ‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ))) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)))
153	78	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑉 ∈ 𝑊)
154	79	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ran (Homf ‘𝐶) ⊆ 𝑈)
155	80	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
156	63	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
157	76	eleq2d 2851	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ℎ ∈ ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) ↔ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)))
158	157	biimpar 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ℎ ∈ ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧))
159	2, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 110, 153, 154, 155, 156, 112, 47, 158	yonedalem4a 18321	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2nd ‘((1st ‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ))))
160	97, 131, 52	natfn 18004	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) Fn 𝐵)
161		dffn5 6929	. . . . . . . . . . 11 ⊢ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ) Fn 𝐵 ↔ ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)))
162	160, 161	sylib 221	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)))
163	152, 159, 162	3eqtr4d 2810	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ) = ((𝑧(2nd ‘𝑌)𝑤)‘ℎ))
164	163	mpteq2dva 5198	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ)) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2nd ‘𝑌)𝑤)‘ℎ)))
165		f1of 6810	. . . . . . . . . 10 ⊢ ((((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))
166	109, 165	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))
167	166	feqmptd 6939	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ)))
168	129	feqmptd 6939	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2nd ‘𝑌)𝑤) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2nd ‘𝑌)𝑤)‘ℎ)))
169	164, 167, 168	3eqtr4d 2810	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑧(2nd ‘𝑌)𝑤))
170	169	f1oeq1d 6805	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)) ↔ (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))))
171	109, 170	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))
172	171	ralrimivva 3208	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))
173	31, 75, 98	isffth2 17965	. . . 4 ⊢ ((1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌) ↔ ((1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤))))
174	16, 172, 173	sylanbrc 594	. . 3 ⊢ (𝜑 → (1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌))
175		df-br 5106	. . 3 ⊢ ((1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌) ↔ ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
176	174, 175	sylib 221	. 2 ⊢ (𝜑 → ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
177	14, 176	eqeltrd 2865	1 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   ∪ cun 3905   ∩ cin 3906   ⊆ wss 3907  ⟨cop 4591   class class class wbr 5105   ↦ cmpt 5186   × cxp 5650  ran crn 5653  Rel wrel 5657   Fn wfn 6520  ⟶wf 6521  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  1^st c1st 7972  2^nd c2nd 7973  tpos ctpos 8209  Basecbs 17259  Hom chom 17311  compcco 17312  Catccat 17710  Idccid 17711  Hom_f chomf 17712  oppCatcoppc 17757  Invcinv 17792  Isociso 17793   Func cfunc 17901   ∘_func ccofu 17903   Full cful 17951   Faith cfth 17952   Nat cnat 17991   FuncCat cfuc 17992  SetCatcsetc 18122   ×_c cxpc 18214   1^st_F c1stf 18215   2^nd_F c2ndf 18216   ⟨,⟩_F cprf 18217   eval_F cevlf 18255  Hom_Fchof 18294  Yoncyon 18295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-tpos 8210  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-pm 8815  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-z 12583  df-dec 12703  df-uz 12854  df-fz 13527  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-hom 17324  df-cco 17325  df-cat 17714  df-cid 17715  df-homf 17716  df-comf 17717  df-oppc 17758  df-sect 17794  df-inv 17795  df-iso 17796  df-ssc 17857  df-resc 17858  df-subc 17859  df-func 17905  df-cofu 17907  df-full 17953  df-fth 17954  df-nat 17993  df-fuc 17994  df-setc 18123  df-xpc 18218  df-1stf 18219  df-2ndf 18220  df-prf 18221  df-evlf 18259  df-curf 18260  df-hof 18296  df-yon 18297

This theorem is referenced by:  yonffth  18330