Theorem yonffthlem 18339

Description: Lemma for yonffth 18341. (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 17913	. . 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 4204	. . . . 5 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
10	7, 9	ssexd 5330	. . . 4 ⊢ (𝜑 → 𝑈 ∈ V)
11		yoneda.u	. . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
12	2, 3, 4, 5, 6, 10, 11	yoncl 18319	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
13		1st2nd 8063	. . 3 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
14	1, 12, 13	sylancr 587	. 2 ⊢ (𝜑 → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
15		1st2ndbr 8066	. . . . 5 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
16	1, 12, 15	sylancr 587	. . . 4 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
17		fveq2 6907	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → (𝑁‘𝑣) = (𝑁‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩))
18		df-ov 7434	. . . . . . . . . . 11 ⊢ (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑁‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩)
19	17, 18	eqtr4di 2793	. . . . . . . . . 10 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → (𝑁‘𝑣) = (((1st ‘𝑌)‘𝑤)𝑁𝑧))
20		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝐸)‘𝑣) = ((1st ‘𝐸)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩))
21		df-ov 7434	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = ((1st ‘𝐸)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩)
22	20, 21	eqtr4di 2793	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝐸)‘𝑣) = (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧))
23		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝑍)‘𝑣) = ((1st ‘𝑍)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩))
24		df-ov 7434	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧) = ((1st ‘𝑍)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩)
25	23, 24	eqtr4di 2793	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝑍)‘𝑣) = (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧))
26	22, 25	oveq12d 7449	. . . . . . . . . 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 2833	. . . . . . . . 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 18016	. . . . . . . . . . . . 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 18329	. . . . . . . . . . . . . . . . 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 17914	. . . . . . . . . . . . . . . 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 18027	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Cat)
44	37	simprd 495	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
45		eqid 2735	. . . . . . . . . . . . 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 18338	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀(𝑍𝐼𝐸)𝑁)
49	29, 30, 43, 38, 44, 45, 48	inviso2 17815	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (𝐸(Iso‘𝑅)𝑍))
50		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
51	6	fucbas 18016	. . . . . . . . . . . . . 14 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
52	4, 31	oppcbas 17764	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝑂)
53	50, 51, 52	xpcbas 18234	. . . . . . . . . . . . 13 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
54		eqid 2735	. . . . . . . . . . . . 13 ⊢ ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
55		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Iso‘𝑇) = (Iso‘𝑇)
56	28, 53, 54, 44, 38, 45, 55	fuciso 18032	. . . . . . . . . . . 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 17917	. . . . . . . . . . . 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 7105	. . . . . . . . . 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 5728	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
66	27, 59, 65	rspcdva 3623	. . . . . . . 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 17769	. . . . . . . . . . . . 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 18139	. . . . . . . . . . . . 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 18277	. . . . . . . . . 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 2735	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
76	2, 31, 74, 62, 75, 64	yon11 18321	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) = (𝑧(Hom ‘𝐶)𝑤))
77	73, 76	eqtrd 2775	. . . . . . . . 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 18330	. . . . . . . . 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 7449	. . . . . . . 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 2841	. . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ (Base‘𝑆) = (Base‘𝑆)
86		relfunc 17913	. . . . . . . . . . . . . 14 ⊢ Rel (𝑂 Func 𝑆)
87		1st2ndbr 8066	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘((1st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑤)))
89	52, 85, 88	funcf1 17917	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘((1st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
90	89, 64	ffvelcdmd 7105	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ (Base‘𝑆))
91	5, 10	setcbas 18132	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
92	91	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 = (Base‘𝑆))
93	90, 92	eleqtrrd 2842	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ 𝑈)
94	76, 93	eqeltrrd 2840	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑈)
95	84, 94	sseldd 3996	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑉)
96		eqid 2735	. . . . . . . . . 10 ⊢ (Homf ‘𝑄) = (Homf ‘𝑄)
97		eqid 2735	. . . . . . . . . . 11 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
98	6, 97	fuchom 18017	. . . . . . . . . 10 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
99	61, 64	ffvelcdmd 7105	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
100	96, 51, 98, 99, 63	homfval 17737	. . . . . . . . 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 4203	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉)
102	101	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ran (Homf ‘𝑄) ⊆ 𝑉)
103	96, 51	homffn 17738	. . . . . . . . . . 11 ⊢ (Homf ‘𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆))
104		fnovrn 7608	. . . . . . . . . . 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 1465	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ ran (Homf ‘𝑄))
106	102, 105	sseldd 3996	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ 𝑉)
107	100, 106	eqeltrrd 2840	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)) ∈ 𝑉)
108	33, 78, 95, 107, 55	setciso 18145	. . . . . . 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 18321	. . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . 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 18322	. . . . . . . . . . . . . 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 18323	. . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 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 5237	. . . . . . . . . . . 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 17919	. . . . . . . . . . . . . . . . . 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 7104	. . . . . . . . . . . . . . . . 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 18006	. . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . 15 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
134	97, 132, 52, 133, 114	natcl 18008	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 ∈ V)
137	60	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆))
138	137, 112	ffvelcdmd 7105	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
139		1st2ndbr 8066	. . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . 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 17917	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st ‘𝑌)‘𝑧)):𝐵⟶(Base‘𝑆))
142	141	ffvelcdmda 7104	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦) ∈ (Base‘𝑆))
143	92	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 = (Base‘𝑆))
144	142, 143	eleqtrrd 2842	. . . . . . . . . . . . . . 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 7104	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑦) ∈ (Base‘𝑆))
147	146, 143	eleqtrrd 2842	. . . . . . . . . . . . . . 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 18135	. . . . . . . . . . . . . 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 6977	. . . . . . . . . . . 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 2778	. . . . . . . . . . 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 5248	. . . . . . . . . 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 2825	. . . . . . . . . . . 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 18332	. . . . . . . . . 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 18009	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) Fn 𝐵)
161		dffn5 6967	. . . . . . . . . . 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 2785	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ) = ((𝑧(2nd ‘𝑌)𝑤)‘ℎ))
164	163	mpteq2dva 5248	. . . . . . . 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 6849	. . . . . . . . . 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 6977	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ)))
168	129	feqmptd 6977	. . . . . . . 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 2785	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑧(2nd ‘𝑌)𝑤))
170	169	f1oeq1d 6844	. . . . . 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 3200	. . . 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 17970	. . . 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 583	. . 3 ⊢ (𝜑 → (1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌))
175		df-br 5149	. . 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 2839	1 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ∪ cun 3961   ∩ cin 3962   ⊆ wss 3963  ⟨cop 4637   class class class wbr 5148   ↦ cmpt 5231   × cxp 5687  ran crn 5690  Rel wrel 5694   Fn wfn 6558  ⟶wf 6559  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  1^st c1st 8011  2^nd c2nd 8012  tpos ctpos 8249  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Idccid 17710  Hom_f chomf 17711  oppCatcoppc 17756  Invcinv 17793  Isociso 17794   Func cfunc 17905   ∘_func ccofu 17907   Full cful 17956   Faith cfth 17957   Nat cnat 17996   FuncCat cfuc 17997  SetCatcsetc 18129   ×_c cxpc 18224   1^st_F c1stf 18225   2^nd_F c2ndf 18226   ⟨,⟩_F cprf 18227   eval_F cevlf 18266  Hom_Fchof 18305  Yoncyon 18306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-hom 17322  df-cco 17323  df-cat 17713  df-cid 17714  df-homf 17715  df-comf 17716  df-oppc 17757  df-sect 17795  df-inv 17796  df-iso 17797  df-ssc 17858  df-resc 17859  df-subc 17860  df-func 17909  df-cofu 17911  df-full 17958  df-fth 17959  df-nat 17998  df-fuc 17999  df-setc 18130  df-xpc 18228  df-1stf 18229  df-2ndf 18230  df-prf 18231  df-evlf 18270  df-curf 18271  df-hof 18307  df-yon 18308

This theorem is referenced by:  yonffth  18341