Theorem yonffthlem 16745

Description: Lemma for yonffth 16747. (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 16345	. . 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 3753	. . . . 5 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
10	7, 9	ssexd 4733	. . . 4 ⊢ (𝜑 → 𝑈 ∈ V)
11		yoneda.u	. . . 4 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
12	2, 3, 4, 5, 6, 10, 11	yoncl 16725	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
13		1st2nd 7105	. . 3 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
14	1, 12, 13	sylancr 694	. 2 ⊢ (𝜑 → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
15		1st2ndbr 7108	. . . . 5 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
16	1, 12, 15	sylancr 694	. . . 4 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
17		yoneda.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐶)
18	6	fucbas 16443	. . . . . . . . . . . . 13 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
19	17, 18, 16	funcf1 16349	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆))
20	19	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆))
21		simprr 792	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
22	20, 21	ffvelrnd 6268	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
23		simprl 790	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
24		opelxpi 5072	. . . . . . . . . 10 ⊢ ((((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆) ∧ 𝑧 ∈ 𝐵) → ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
25	22, 23, 24	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
26		yoneda.r	. . . . . . . . . . . . . 14 ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
27	26	fucbas 16443	. . . . . . . . . . . . 13 ⊢ ((𝑄 ×c 𝑂) Func 𝑇) = (Base‘𝑅)
28		yonedainv.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (Inv‘𝑅)
29		yoneda.1	. . . . . . . . . . . . . . . . . 18 ⊢ 1 = (Id‘𝐶)
30		yoneda.t	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑇 = (SetCat‘𝑉)
31		yoneda.h	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐻 = (HomF‘𝑄)
32		yoneda.e	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐸 = (𝑂 evalF 𝑆)
33		yoneda.z	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑍 = (𝐻 ∘func ((⟨(1st ‘𝑌), tpos (2nd ‘𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
34	2, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 3, 7, 11, 8	yonedalem1 16735	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)))
35	34	simpld 474	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
36		funcrcl 16346	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
37	35, 36	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑄 ×c 𝑂) ∈ Cat ∧ 𝑇 ∈ Cat))
38	37	simpld 474	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄 ×c 𝑂) ∈ Cat)
39	37	simprd 478	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ∈ Cat)
40	26, 38, 39	fuccat 16453	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Cat)
41	34	simprd 478	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))
42		eqid 2610	. . . . . . . . . . . . 13 ⊢ (Iso‘𝑅) = (Iso‘𝑅)
43		yoneda.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))))
44		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)))))
45	2, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 3, 7, 11, 8, 43, 28, 44	yonedainv 16744	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀(𝑍𝐼𝐸)𝑁)
46	27, 28, 40, 35, 41, 42, 45	inviso2 16250	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (𝐸(Iso‘𝑅)𝑍))
47		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂)
48	4, 17	oppcbas 16201	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝑂)
49	47, 18, 48	xpcbas 16641	. . . . . . . . . . . . 13 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×c 𝑂))
50		eqid 2610	. . . . . . . . . . . . 13 ⊢ ((𝑄 ×c 𝑂) Nat 𝑇) = ((𝑄 ×c 𝑂) Nat 𝑇)
51		eqid 2610	. . . . . . . . . . . . 13 ⊢ (Iso‘𝑇) = (Iso‘𝑇)
52	26, 49, 50, 41, 35, 42, 51	fuciso 16458	. . . . . . . . . . . 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)))))
53	46, 52	mpbid 221	. . . . . . . . . . 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))))
54	53	simprd 478	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)))
55	54	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1st ‘𝐸)‘𝑣)(Iso‘𝑇)((1st ‘𝑍)‘𝑣)))
56		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → (𝑁‘𝑣) = (𝑁‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩))
57		df-ov 6552	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑁‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩)
58	56, 57	syl6eqr 2662	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → (𝑁‘𝑣) = (((1st ‘𝑌)‘𝑤)𝑁𝑧))
59		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝐸)‘𝑣) = ((1st ‘𝐸)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩))
60		df-ov 6552	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = ((1st ‘𝐸)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩)
61	59, 60	syl6eqr 2662	. . . . . . . . . . . 12 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝐸)‘𝑣) = (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧))
62		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝑍)‘𝑣) = ((1st ‘𝑍)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩))
63		df-ov 6552	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧) = ((1st ‘𝑍)‘⟨((1st ‘𝑌)‘𝑤), 𝑧⟩)
64	62, 63	syl6eqr 2662	. . . . . . . . . . . 12 ⊢ (𝑣 = ⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ → ((1st ‘𝑍)‘𝑣) = (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧))
65	61, 64	oveq12d 6567	. . . . . . . . . . 11 ⊢ (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)))
66	58, 65	eleq12d 2682	. . . . . . . . . 10 ⊢ (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))))
67	66	rspcv 3278	. . . . . . . . 9 ⊢ (⟨((1st ‘𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵) → (∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(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))))
68	25, 55, 67	sylc 63	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧)(Iso‘𝑇)(((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧)))
69	4	oppccat 16205	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
70	3, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂 ∈ Cat)
71	70	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑂 ∈ Cat)
72	5	setccat 16558	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
73	10, 72	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Cat)
74	73	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑆 ∈ Cat)
75	32, 71, 74, 48, 22, 23	evlf1 16683	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧))
76	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝐶 ∈ Cat)
77		eqid 2610	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
78	2, 17, 76, 21, 77, 23	yon11 16727	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) = (𝑧(Hom ‘𝐶)𝑤))
79	75, 78	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝐸)𝑧) = (𝑧(Hom ‘𝐶)𝑤))
80	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑉 ∈ 𝑊)
81	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ran (Homf ‘𝐶) ⊆ 𝑈)
82	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
83	2, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 76, 80, 81, 82, 22, 23	yonedalem21 16736	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)(1st ‘𝑍)𝑧) = (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))
84	79, 83	oveq12d 6567	. . . . . . . 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))))
85	68, 84	eleqtrd 2690	. . . . . . 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))))
86	9	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 ⊆ 𝑉)
87		eqid 2610	. . . . . . . . . . . . 13 ⊢ (Base‘𝑆) = (Base‘𝑆)
88		relfunc 16345	. . . . . . . . . . . . . 14 ⊢ Rel (𝑂 Func 𝑆)
89		1st2ndbr 7108	. . . . . . . . . . . . . 14 ⊢ ((Rel (𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑤)))
90	88, 22, 89	sylancr 694	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘((1st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑤)))
91	48, 87, 90	funcf1 16349	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘((1st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
92	91, 23	ffvelrnd 6268	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ (Base‘𝑆))
93	5, 10	setcbas 16551	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
94	93	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 = (Base‘𝑆))
95	92, 94	eleqtrrd 2691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) ∈ 𝑈)
96	78, 95	eqeltrrd 2689	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑈)
97	86, 96	sseldd 3569	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑉)
98		eqid 2610	. . . . . . . . . 10 ⊢ (Homf ‘𝑄) = (Homf ‘𝑄)
99		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
100	6, 99	fuchom 16444	. . . . . . . . . 10 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
101	20, 23	ffvelrnd 6268	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
102	98, 18, 100, 101, 22	homfval 16175	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) = (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))
103	8	unssad 3752	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉)
104	103	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ran (Homf ‘𝑄) ⊆ 𝑉)
105	98, 18	homffn 16176	. . . . . . . . . . . 12 ⊢ (Homf ‘𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆))
106	105	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (Homf ‘𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆)))
107		fnovrn 6707	. . . . . . . . . . 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))
108	106, 101, 22, 107	syl3anc 1318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ ran (Homf ‘𝑄))
109	104, 108	sseldd 3569	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(Homf ‘𝑄)((1st ‘𝑌)‘𝑤)) ∈ 𝑉)
110	102, 109	eqeltrrd 2689	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)) ∈ 𝑉)
111	30, 80, 97, 110, 51	setciso 16564	. . . . . . 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))))
112	85, 111	mpbid 221	. . . . . 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)))
113	76	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝐶 ∈ Cat)
114	113	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
115	23	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑧 ∈ 𝐵)
116	115	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑧 ∈ 𝐵)
117		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
118	2, 17, 114, 116, 77, 117	yon11 16727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦) = (𝑦(Hom ‘𝐶)𝑧))
119	118	eqcomd 2616	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (𝑦(Hom ‘𝐶)𝑧) = ((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦))
120	114	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝐶 ∈ Cat)
121	21	ad3antrrr 762	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑤 ∈ 𝐵)
122	116	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑧 ∈ 𝐵)
123		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (comp‘𝐶) = (comp‘𝐶)
124	117	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑦 ∈ 𝐵)
125		simpr 476	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
126		simpllr 795	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ℎ ∈ (𝑧(Hom ‘𝐶)𝑤))
127	2, 17, 120, 121, 77, 122, 123, 124, 125, 126	yon12 16728	. . . . . . . . . . . . . 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))
128	2, 17, 120, 122, 77, 121, 123, 124, 126, 125	yon2 16729	. . . . . . . . . . . . . 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))
129	127, 128	eqtr4d 2647	. . . . . . . . . . . . 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))
130	119, 129	mpteq12dva 4662	. . . . . . . . . . . 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)))
131	16	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌))
132	17, 77, 100, 131, 23, 21	funcf2 16351	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)⟶(((1st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1st ‘𝑌)‘𝑤)))
133	132	ffvelrnda 6267	. . . . . . . . . . . . . . . . 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)))
134	99, 133	nat1st2nd 16434	. . . . . . . . . . . . . . . 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))⟩))
135	134	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))⟩))
136		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
137	99, 135, 48, 136, 117	natcl 16436	. . . . . . . . . . . . . 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)))
138	10	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 ∈ V)
139	138	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 ∈ V)
140	19	ad2antrr 758	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘𝑌):𝐵⟶(𝑂 Func 𝑆))
141	140, 115	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
142		1st2ndbr 7108	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Rel (𝑂 Func 𝑆) ∧ ((1st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st ‘𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑧)))
143	88, 141, 142	sylancr 694	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st ‘𝑌)‘𝑧))(𝑂 Func 𝑆)(2nd ‘((1st ‘𝑌)‘𝑧)))
144	48, 87, 143	funcf1 16349	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st ‘𝑌)‘𝑧)):𝐵⟶(Base‘𝑆))
145	144	ffvelrnda 6267	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦) ∈ (Base‘𝑆))
146	94	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 = (Base‘𝑆))
147	145, 146	eleqtrrd 2691	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑧))‘𝑦) ∈ 𝑈)
148	91	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1st ‘((1st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
149	148	ffvelrnda 6267	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑦) ∈ (Base‘𝑆))
150	149, 146	eleqtrrd 2691	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑦) ∈ 𝑈)
151	5, 139, 136, 147, 150	elsetchom 16554	. . . . . . . . . . . . . 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)))
152	137, 151	mpbid 221	. . . . . . . . . . . . 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))
153	152	feqmptd 6159	. . . . . . . . . . . 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)))
154	130, 153	eqtr4d 2647	. . . . . . . . . . 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))
155	154	mpteq2dva 4672	. . . . . . . . . 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)))
156	80	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑉 ∈ 𝑊)
157	81	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ran (Homf ‘𝐶) ⊆ 𝑈)
158	82	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
159	22	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
160	78	eleq2d 2673	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ℎ ∈ ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧) ↔ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)))
161	160	biimpar 501	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ℎ ∈ ((1st ‘((1st ‘𝑌)‘𝑤))‘𝑧))
162	2, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 113, 156, 157, 158, 159, 115, 44, 161	yonedalem4a 16738	. . . . . . . . . 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))))
163	99, 134, 48	natfn 16437	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) Fn 𝐵)
164		dffn5 6151	. . . . . . . . . . 11 ⊢ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ) Fn 𝐵 ↔ ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)))
165	163, 164	sylib 207	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2nd ‘𝑌)𝑤)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2nd ‘𝑌)𝑤)‘ℎ)‘𝑦)))
166	155, 162, 165	3eqtr4d 2654	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ) = ((𝑧(2nd ‘𝑌)𝑤)‘ℎ))
167	166	mpteq2dva 4672	. . . . . . . 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)))
168		f1of 6050	. . . . . . . . . 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)))
169	112, 168	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)))
170	169	feqmptd 6159	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ)))
171	132	feqmptd 6159	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2nd ‘𝑌)𝑤) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2nd ‘𝑌)𝑤)‘ℎ)))
172	167, 170, 171	3eqtr4d 2654	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑧(2nd ‘𝑌)𝑤))
173		f1oeq1 6040	. . . . . . 7 ⊢ ((((1st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑧(2nd ‘𝑌)𝑤) → ((((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))))
174	172, 173	syl 17	. . . . . 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))))
175	112, 174	mpbid 221	. . . . 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)))
176	175	ralrimivva 2954	. . . 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)))
177	17, 77, 100	isffth2 16399	. . . 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))))
178	16, 176, 177	sylanbrc 695	. . 3 ⊢ (𝜑 → (1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌))
179		df-br 4584	. . 3 ⊢ ((1st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2nd ‘𝑌) ↔ ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
180	178, 179	sylib 207	. 2 ⊢ (𝜑 → ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
181	14, 180	eqeltrd 2688	1 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ran crn 5039  Rel wrel 5043   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058  tpos ctpos 7238  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149  Hom_f chomf 16150  oppCatcoppc 16194  Invcinv 16228  Isociso 16229   Func cfunc 16337   ∘_func ccofu 16339   Full cful 16385   Faith cfth 16386   Nat cnat 16424   FuncCat cfuc 16425  SetCatcsetc 16548   ×_c cxpc 16631   1^st_F c1stf 16632   2^nd_F c2ndf 16633   ⟨,⟩_F cprf 16634   eval_F cevlf 16672  Hom_Fchof 16711  Yoncyon 16712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-homf 16154  df-comf 16155  df-oppc 16195  df-sect 16230  df-inv 16231  df-iso 16232  df-ssc 16293  df-resc 16294  df-subc 16295  df-func 16341  df-cofu 16343  df-full 16387  df-fth 16388  df-nat 16426  df-fuc 16427  df-setc 16549  df-xpc 16635  df-1stf 16636  df-2ndf 16637  df-prf 16638  df-evlf 16676  df-curf 16677  df-hof 16713  df-yon 16714

This theorem is referenced by:  yonffth  16747