Theorem yonffthlem 18217

Description: Lemma for yonffth 18219. (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 17798	. . 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 4148	. . . . 5 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
10	7, 9	ssexd 5271	. . . 4 ⊢ (𝜑 → 𝑈 ∈ V)
11		yoneda.u	. . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
12	2, 3, 4, 5, 6, 10, 11	yoncl 18197	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
13		1st2nd 7993	. . 3 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
14	1, 12, 13	sylancr 588	. 2 ⊢ (𝜑 → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
15		1st2ndbr 7996	. . . . 5 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
16	1, 12, 15	sylancr 588	. . . 4 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
17		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → (𝑁‘𝑣) = (𝑁‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩))
18		df-ov 7371	. . . . . . . . . . 11 ⊢ (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑁‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩)
19	17, 18	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → (𝑁‘𝑣) = (((1st ‘𝑌)‘𝑤)𝑁𝑧))
20		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝐸)‘𝑣) = ((1st ‘𝐸)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩))
21		df-ov 7371	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = ((1st ‘𝐸)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩)
22	20, 21	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝐸)‘𝑣) = (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧))
23		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝑍)‘𝑣) = ((1st ‘𝑍)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩))
24		df-ov 7371	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧) = ((1st ‘𝑍)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩)
25	23, 24	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝑍)‘𝑣) = (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧))
26	22, 25	oveq12d 7386	. . . . . . . . . 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 2831	. . . . . . . . 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 17899	. . . . . . . . . . . . 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 18207	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
38	37	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
39		funcrcl 17799	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
40	38, 39	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
41	40	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄 ×c 𝑂) ∈ Cat)
42	40	simprd 495	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ∈ Cat)
43	28, 41, 42	fuccat 17909	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Cat)
44	37	simprd 495	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
45		eqid 2737	. . . . . . . . . . . . 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 18216	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀(𝑍𝐼𝐸)𝑁)
49	29, 30, 43, 38, 44, 45, 48	inviso2 17703	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (𝐸(Iso‘𝑅)𝑍))
50		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
51	6	fucbas 17899	. . . . . . . . . . . . . 14 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
52	4, 31	oppcbas 17653	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝑂)
53	50, 51, 52	xpcbas 18113	. . . . . . . . . . . . 13 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
54		eqid 2737	. . . . . . . . . . . . 13 ⊢ ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
55		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Iso‘𝑇) = (Iso‘𝑇)
56	28, 53, 54, 44, 38, 45, 55	fuciso 17914	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 495	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)))
59	58	adantr 480	. . . . . . . . 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 17802	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆))
61	60	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆))
62		simprr 773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
63	61, 62	ffvelcdmd 7039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
64		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
65	63, 64	opelxpd 5671	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
66	27, 59, 65	rspcdva 3579	. . . . . . . 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 17657	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
68	3, 67	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂 ∈ Cat)
69	68	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑂 ∈ Cat)
70	5	setccat 18021	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
71	10, 70	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Cat)
72	71	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑆 ∈ Cat)
73	35, 69, 72, 52, 63, 64	evlf1 18155	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧))
74	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝐶 ∈ Cat)
75		eqid 2737	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
76	2, 31, 74, 62, 75, 64	yon11 18199	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) = (𝑧(Hom ‘𝐶)𝑤))
77	73, 76	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = (𝑧(Hom ‘𝐶)𝑤))
78	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑉 ∈ 𝑊)
79	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ran (Homf ‘𝐶) ⊆ 𝑈)
80	8	adantr 480	. . . . . . . . . 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 18208	. . . . . . . . 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 7386	. . . . . . . 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 2839	. . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 ⊆ 𝑉)
85		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Base‘𝑆) = (Base‘𝑆)
86		relfunc 17798	. . . . . . . . . . . . . 14 ⊢ Rel (𝑂 Func 𝑆)
87		1st2ndbr 7996	. . . . . . . . . . . . . 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 588	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘((1st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑤)))
89	52, 85, 88	funcf1 17802	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘((1st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
90	89, 64	ffvelcdmd 7039	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ (Base‘𝑆))
91	5, 10	setcbas 18014	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
92	91	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 = (Base‘𝑆))
93	90, 92	eleqtrrd 2840	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ 𝑈)
94	76, 93	eqeltrrd 2838	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑈)
95	84, 94	sseldd 3936	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑉)
96		eqid 2737	. . . . . . . . . 10 ⊢ (Homf ‘𝑄) = (Homf ‘𝑄)
97		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
98	6, 97	fuchom 17900	. . . . . . . . . 10 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
99	61, 64	ffvelcdmd 7039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
100	96, 51, 98, 99, 63	homfval 17627	. . . . . . . . 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 4147	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉)
102	101	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ran (Homf ‘𝑄) ⊆ 𝑉)
103	96, 51	homffn 17628	. . . . . . . . . . 11 ⊢ (Homf ‘𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆))
104		fnovrn 7543	. . . . . . . . . . 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 1469	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ ran (Homf ‘𝑄))
106	102, 105	sseldd 3936	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ 𝑉)
107	100, 106	eqeltrrd 2838	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)) ∈ 𝑉)
108	33, 78, 95, 107, 55	setciso 18027	. . . . . . 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 232	. . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝐶 ∈ Cat)
111	110	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
112	64	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑧 ∈ 𝐵)
113	112	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑧 ∈ 𝐵)
114		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
115	2, 31, 111, 113, 75, 114	yon11 18199	. . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝐶 ∈ Cat)
118	62	ad3antrrr 731	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑤 ∈ 𝐵)
119	113	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑧 ∈ 𝐵)
120		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (comp‘𝐶) = (comp‘𝐶)
121	114	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑦 ∈ 𝐵)
122		simpr 484	. . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . 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 18200	. . . . . . . . . . . . . 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 18201	. . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . 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 5186	. . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
129	31, 75, 98, 128, 64, 62	funcf2 17804	. . . . . . . . . . . . . . . . . 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 7038	. . . . . . . . . . . . . . . . 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 17890	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
134	97, 132, 52, 133, 114	natcl 17892	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 ∈ V)
136	135	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 ∈ V)
137	60	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆))
138	137, 112	ffvelcdmd 7039	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
139		1st2ndbr 7996	. . . . . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . . . . 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 17802	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st ‘𝑌)‘𝑧)):𝐵⟶(Base‘𝑆))
142	141	ffvelcdmda 7038	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦) ∈ (Base‘𝑆))
143	92	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 = (Base‘𝑆))
144	142, 143	eleqtrrd 2840	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦) ∈ 𝑈)
145	89	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
146	145	ffvelcdmda 7038	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑦) ∈ (Base‘𝑆))
147	146, 143	eleqtrrd 2840	. . . . . . . . . . . . . . 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 18017	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 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 6910	. . . . . . . . . . . 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 2775	. . . . . . . . . . 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 5193	. . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑉 ∈ 𝑊)
154	79	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ran (Homf ‘𝐶) ⊆ 𝑈)
155	80	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
156	63	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
157	76	eleq2d 2823	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ℎ ∈ ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) ↔ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)))
158	157	biimpar 477	. . . . . . . . . . 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 18210	. . . . . . . . . 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 17893	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) Fn 𝐵)
161		dffn5 6900	. . . . . . . . . . 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 218	. . . . . . . . . 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 2782	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ) = ((𝑧(2nd ‘𝑌)𝑤)‘ℎ))
164	163	mpteq2dva 5193	. . . . . . . 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 6782	. . . . . . . . . 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 17	. . . . . . . . 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 6910	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ)))
168	129	feqmptd 6910	. . . . . . . 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 2782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑧(2nd ‘𝑌)𝑤))
170	169	f1oeq1d 6777	. . . . . 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 232	. . . . 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 3181	. . . 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 17854	. . . 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 584	. . 3 ⊢ (𝜑 → (1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌))
175		df-br 5101	. . 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 218	. 2 ⊢ (𝜑 → ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
177	14, 176	eqeltrd 2837	1 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ⟨cop 4588   class class class wbr 5100   ↦ cmpt 5181   × cxp 5630  ran crn 5633  Rel wrel 5637   Fn wfn 6495  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1^st c1st 7941  2^nd c2nd 7942  tpos ctpos 8177  Basecbs 17148  Hom chom 17200  compcco 17201  Catccat 17599  Idccid 17600  Hom_f chomf 17601  oppCatcoppc 17646  Invcinv 17681  Isociso 17682   Func cfunc 17790   ∘_func ccofu 17792   Full cful 17840   Faith cfth 17841   Nat cnat 17880   FuncCat cfuc 17881  SetCatcsetc 18011   ×_c cxpc 18103   1^st_F c1stf 18104   2^nd_F c2ndf 18105   ⟨,⟩_F cprf 18106   eval_F cevlf 18144  Hom_Fchof 18183  Yoncyon 18184

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-tpos 8178  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-pm 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-hom 17213  df-cco 17214  df-cat 17603  df-cid 17604  df-homf 17605  df-comf 17606  df-oppc 17647  df-sect 17683  df-inv 17684  df-iso 17685  df-ssc 17746  df-resc 17747  df-subc 17748  df-func 17794  df-cofu 17796  df-full 17842  df-fth 17843  df-nat 17882  df-fuc 17883  df-setc 18012  df-xpc 18107  df-1stf 18108  df-2ndf 18109  df-prf 18110  df-evlf 18148  df-curf 18149  df-hof 18185  df-yon 18186

This theorem is referenced by:  yonffth  18219