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Theorem yonffthlem 18232

Description: Lemma for yonffth 18234. (Contributed by Mario Carneiro, 29-Jan-2017.)

**Hypotheses**
Ref	Expression
yoneda.y	⊢ 𝑌 = (Yon‘𝐶)
yoneda.b	⊢ 𝐵 = (Base‘𝐶)
yoneda.1	⊢ 1 = (Id‘𝐶)
yoneda.o	⊢ 𝑂 = (oppCat‘𝐶)
yoneda.s	⊢ 𝑆 = (SetCat‘𝑈)
yoneda.t	⊢ 𝑇 = (SetCat‘𝑉)
yoneda.q	⊢ 𝑄 = (𝑂 FuncCat 𝑆)
yoneda.h	⊢ 𝐻 = (Hom_F‘𝑄)
yoneda.r	⊢ 𝑅 = ((𝑄 ×_c 𝑂) FuncCat 𝑇)
yoneda.e	⊢ 𝐸 = (𝑂 eval_F 𝑆)
yoneda.z	⊢ 𝑍 = (𝐻 ∘_func ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)))
yoneda.c	⊢ (𝜑 → 𝐶 ∈ Cat)
yoneda.w	⊢ (𝜑 → 𝑉 ∈ 𝑊)
yoneda.u	⊢ (𝜑 → ran (Hom_f ‘𝐶) ⊆ 𝑈)
yoneda.v	⊢ (𝜑 → (ran (Hom_f ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
yoneda.m	⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1^st ‘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 ∈ ((1^st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2^nd ‘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 17809	. . 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 (Hom_f ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
9	8	unssbd 4188	. . . . 5 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
10	7, 9	ssexd 5324	. . . 4 ⊢ (𝜑 → 𝑈 ∈ V)
11		yoneda.u	. . . 4 ⊢ (𝜑 → ran (Hom_f ‘𝐶) ⊆ 𝑈)
12	2, 3, 4, 5, 6, 10, 11	yoncl 18212	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄))
13		1st2nd 8022	. . 3 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → 𝑌 = ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)
14	1, 12, 13	sylancr 588	. 2 ⊢ (𝜑 → 𝑌 = ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩)
15		1st2ndbr 8025	. . . . 5 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌))
16	1, 12, 15	sylancr 588	. . . 4 ⊢ (𝜑 → (1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌))
17		fveq2 6889	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩ → (𝑁‘𝑣) = (𝑁‘⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩))
18		df-ov 7409	. . . . . . . . . . 11 ⊢ (((1^st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑁‘⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩)
19	17, 18	eqtr4di 2791	. . . . . . . . . 10 ⊢ (𝑣 = ⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩ → (𝑁‘𝑣) = (((1^st ‘𝑌)‘𝑤)𝑁𝑧))
20		fveq2 6889	. . . . . . . . . . . 12 ⊢ (𝑣 = ⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩ → ((1^st ‘𝐸)‘𝑣) = ((1^st ‘𝐸)‘⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩))
21		df-ov 7409	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑌)‘𝑤)(1^st ‘𝐸)𝑧) = ((1^st ‘𝐸)‘⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩)
22	20, 21	eqtr4di 2791	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩ → ((1^st ‘𝐸)‘𝑣) = (((1^st ‘𝑌)‘𝑤)(1^st ‘𝐸)𝑧))
23		fveq2 6889	. . . . . . . . . . . 12 ⊢ (𝑣 = ⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩ → ((1^st ‘𝑍)‘𝑣) = ((1^st ‘𝑍)‘⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩))
24		df-ov 7409	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑌)‘𝑤)(1^st ‘𝑍)𝑧) = ((1^st ‘𝑍)‘⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩)
25	23, 24	eqtr4di 2791	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩ → ((1^st ‘𝑍)‘𝑣) = (((1^st ‘𝑌)‘𝑤)(1^st ‘𝑍)𝑧))
26	22, 25	oveq12d 7424	. . . . . . . . . 10 ⊢ (𝑣 = ⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩ → (((1^st ‘𝐸)‘𝑣)(Iso‘𝑇)((1^st ‘𝑍)‘𝑣)) = ((((1^st ‘𝑌)‘𝑤)(1^st ‘𝐸)𝑧)(Iso‘𝑇)(((1^st ‘𝑌)‘𝑤)(1^st ‘𝑍)𝑧)))
27	19, 26	eleq12d 2828	. . . . . . . . 9 ⊢ (𝑣 = ⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩ → ((𝑁‘𝑣) ∈ (((1^st ‘𝐸)‘𝑣)(Iso‘𝑇)((1^st ‘𝑍)‘𝑣)) ↔ (((1^st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((((1^st ‘𝑌)‘𝑤)(1^st ‘𝐸)𝑧)(Iso‘𝑇)(((1^st ‘𝑌)‘𝑤)(1^st ‘𝑍)𝑧))))
28		yoneda.r	. . . . . . . . . . . . . 14 ⊢ 𝑅 = ((𝑄 ×_c 𝑂) FuncCat 𝑇)
29	28	fucbas 17909	. . . . . . . . . . . . 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 = (Hom_F‘𝑄)
35		yoneda.e	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐸 = (𝑂 eval_F 𝑆)
36		yoneda.z	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑍 = (𝐻 ∘_func ((⟨(1^st ‘𝑌), tpos (2^nd ‘𝑌)⟩ ∘_func (𝑄 2^nd_F 𝑂)) ⟨,⟩_F (𝑄 1^st_F 𝑂)))
37	2, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 3, 7, 11, 8	yonedalem1 18222	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×_c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×_c 𝑂) Func 𝑇)))
38	37	simpld 496	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×_c 𝑂) Func 𝑇))
39		funcrcl 17810	. . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄 ×_c 𝑂) ∈ Cat)
42	40	simprd 497	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ∈ Cat)
43	28, 41, 42	fuccat 17920	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ Cat)
44	37	simprd 497	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×_c 𝑂) Func 𝑇))
45		eqid 2733	. . . . . . . . . . . . 13 ⊢ (Iso‘𝑅) = (Iso‘𝑅)
46		yoneda.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑎 ∈ (((1^st ‘𝑌)‘𝑥)(𝑂 Nat 𝑆)𝑓) ↦ ((𝑎‘𝑥)‘( 1 ‘𝑥))))
47		yonedainv.n	. . . . . . . . . . . . . 14 ⊢ 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥 ∈ 𝐵 ↦ (𝑢 ∈ ((1^st ‘𝑓)‘𝑥) ↦ (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2^nd ‘𝑓)𝑦)‘𝑔)‘𝑢)))))
48	2, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 3, 7, 11, 8, 46, 30, 47	yonedainv 18231	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀(𝑍𝐼𝐸)𝑁)
49	29, 30, 43, 38, 44, 45, 48	inviso2 17711	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (𝐸(Iso‘𝑅)𝑍))
50		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (𝑄 ×_c 𝑂) = (𝑄 ×_c 𝑂)
51	6	fucbas 17909	. . . . . . . . . . . . . 14 ⊢ (𝑂 Func 𝑆) = (Base‘𝑄)
52	4, 31	oppcbas 17660	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝑂)
53	50, 51, 52	xpcbas 18127	. . . . . . . . . . . . 13 ⊢ ((𝑂 Func 𝑆) × 𝐵) = (Base‘(𝑄 ×_c 𝑂))
54		eqid 2733	. . . . . . . . . . . . 13 ⊢ ((𝑄 ×_c 𝑂) Nat 𝑇) = ((𝑄 ×_c 𝑂) Nat 𝑇)
55		eqid 2733	. . . . . . . . . . . . 13 ⊢ (Iso‘𝑇) = (Iso‘𝑇)
56	28, 53, 54, 44, 38, 45, 55	fuciso 17925	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 ∈ (𝐸(Iso‘𝑅)𝑍) ↔ (𝑁 ∈ (𝐸((𝑄 ×_c 𝑂) Nat 𝑇)𝑍) ∧ ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1^st ‘𝐸)‘𝑣)(Iso‘𝑇)((1^st ‘𝑍)‘𝑣)))))
57	49, 56	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∈ (𝐸((𝑄 ×_c 𝑂) Nat 𝑇)𝑍) ∧ ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1^st ‘𝐸)‘𝑣)(Iso‘𝑇)((1^st ‘𝑍)‘𝑣))))
58	57	simprd 497	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1^st ‘𝐸)‘𝑣)(Iso‘𝑇)((1^st ‘𝑍)‘𝑣)))
59	58	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ∀𝑣 ∈ ((𝑂 Func 𝑆) × 𝐵)(𝑁‘𝑣) ∈ (((1^st ‘𝐸)‘𝑣)(Iso‘𝑇)((1^st ‘𝑍)‘𝑣)))
60	31, 51, 16	funcf1 17813	. . . . . . . . . . . 12 ⊢ (𝜑 → (1^st ‘𝑌):𝐵⟶(𝑂 Func 𝑆))
61	60	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1^st ‘𝑌):𝐵⟶(𝑂 Func 𝑆))
62		simprr 772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑤 ∈ 𝐵)
63	61, 62	ffvelcdmd 7085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1^st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
64		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑧 ∈ 𝐵)
65	63, 64	opelxpd 5714	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ⟨((1^st ‘𝑌)‘𝑤), 𝑧⟩ ∈ ((𝑂 Func 𝑆) × 𝐵))
66	27, 59, 65	rspcdva 3614	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1^st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((((1^st ‘𝑌)‘𝑤)(1^st ‘𝐸)𝑧)(Iso‘𝑇)(((1^st ‘𝑌)‘𝑤)(1^st ‘𝑍)𝑧)))
67	4	oppccat 17665	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
68	3, 67	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑂 ∈ Cat)
69	68	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑂 ∈ Cat)
70	5	setccat 18032	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat)
71	10, 70	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Cat)
72	71	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑆 ∈ Cat)
73	35, 69, 72, 52, 63, 64	evlf1 18170	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1^st ‘𝑌)‘𝑤)(1^st ‘𝐸)𝑧) = ((1^st ‘((1^st ‘𝑌)‘𝑤))‘𝑧))
74	3	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝐶 ∈ Cat)
75		eqid 2733	. . . . . . . . . . 11 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
76	2, 31, 74, 62, 75, 64	yon11 18214	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1^st ‘((1^st ‘𝑌)‘𝑤))‘𝑧) = (𝑧(Hom ‘𝐶)𝑤))
77	73, 76	eqtrd 2773	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1^st ‘𝑌)‘𝑤)(1^st ‘𝐸)𝑧) = (𝑧(Hom ‘𝐶)𝑤))
78	7	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑉 ∈ 𝑊)
79	11	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ran (Hom_f ‘𝐶) ⊆ 𝑈)
80	8	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ran (Hom_f ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
81	2, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 74, 78, 79, 80, 63, 64	yonedalem21 18223	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1^st ‘𝑌)‘𝑤)(1^st ‘𝑍)𝑧) = (((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤)))
82	77, 81	oveq12d 7424	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((1^st ‘𝑌)‘𝑤)(1^st ‘𝐸)𝑧)(Iso‘𝑇)(((1^st ‘𝑌)‘𝑤)(1^st ‘𝑍)𝑧)) = ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤))))
83	66, 82	eleqtrd 2836	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1^st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤))))
84	9	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 ⊆ 𝑉)
85		eqid 2733	. . . . . . . . . . . . 13 ⊢ (Base‘𝑆) = (Base‘𝑆)
86		relfunc 17809	. . . . . . . . . . . . . 14 ⊢ Rel (𝑂 Func 𝑆)
87		1st2ndbr 8025	. . . . . . . . . . . . . 14 ⊢ ((Rel (𝑂 Func 𝑆) ∧ ((1^st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (1^st ‘((1^st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2^nd ‘((1^st ‘𝑌)‘𝑤)))
88	86, 63, 87	sylancr 588	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1^st ‘((1^st ‘𝑌)‘𝑤))(𝑂 Func 𝑆)(2^nd ‘((1^st ‘𝑌)‘𝑤)))
89	52, 85, 88	funcf1 17813	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1^st ‘((1^st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
90	89, 64	ffvelcdmd 7085	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1^st ‘((1^st ‘𝑌)‘𝑤))‘𝑧) ∈ (Base‘𝑆))
91	5, 10	setcbas 18025	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 = (Base‘𝑆))
92	91	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 = (Base‘𝑆))
93	90, 92	eleqtrrd 2837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1^st ‘((1^st ‘𝑌)‘𝑤))‘𝑧) ∈ 𝑈)
94	76, 93	eqeltrrd 2835	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑈)
95	84, 94	sseldd 3983	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(Hom ‘𝐶)𝑤) ∈ 𝑉)
96		eqid 2733	. . . . . . . . . 10 ⊢ (Hom_f ‘𝑄) = (Hom_f ‘𝑄)
97		eqid 2733	. . . . . . . . . . 11 ⊢ (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
98	6, 97	fuchom 17910	. . . . . . . . . 10 ⊢ (𝑂 Nat 𝑆) = (Hom ‘𝑄)
99	61, 64	ffvelcdmd 7085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((1^st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
100	96, 51, 98, 99, 63	homfval 17633	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1^st ‘𝑌)‘𝑧)(Hom_f ‘𝑄)((1^st ‘𝑌)‘𝑤)) = (((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤)))
101	8	unssad 4187	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Hom_f ‘𝑄) ⊆ 𝑉)
102	101	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ran (Hom_f ‘𝑄) ⊆ 𝑉)
103	96, 51	homffn 17634	. . . . . . . . . . 11 ⊢ (Hom_f ‘𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆))
104		fnovrn 7579	. . . . . . . . . . 11 ⊢ (((Hom_f ‘𝑄) Fn ((𝑂 Func 𝑆) × (𝑂 Func 𝑆)) ∧ ((1^st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆) ∧ ((1^st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆)) → (((1^st ‘𝑌)‘𝑧)(Hom_f ‘𝑄)((1^st ‘𝑌)‘𝑤)) ∈ ran (Hom_f ‘𝑄))
105	103, 99, 63, 104	mp3an2i 1467	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1^st ‘𝑌)‘𝑧)(Hom_f ‘𝑄)((1^st ‘𝑌)‘𝑤)) ∈ ran (Hom_f ‘𝑄))
106	102, 105	sseldd 3983	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1^st ‘𝑌)‘𝑧)(Hom_f ‘𝑄)((1^st ‘𝑌)‘𝑤)) ∈ 𝑉)
107	100, 106	eqeltrrd 2835	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤)) ∈ 𝑉)
108	33, 78, 95, 107, 55	setciso 18038	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((1^st ‘𝑌)‘𝑤)𝑁𝑧) ∈ ((𝑧(Hom ‘𝐶)𝑤)(Iso‘𝑇)(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤))) ↔ (((1^st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤))))
109	83, 108	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1^st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤)))
110	74	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝐶 ∈ Cat)
111	110	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
112	64	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑧 ∈ 𝐵)
113	112	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑧 ∈ 𝐵)
114		simpr 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
115	2, 31, 111, 113, 75, 114	yon11 18214	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1^st ‘((1^st ‘𝑌)‘𝑧))‘𝑦) = (𝑦(Hom ‘𝐶)𝑧))
116	115	eqcomd 2739	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (𝑦(Hom ‘𝐶)𝑧) = ((1^st ‘((1^st ‘𝑌)‘𝑧))‘𝑦))
117	111	adantr 482	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝐶 ∈ Cat)
118	62	ad3antrrr 729	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑤 ∈ 𝐵)
119	113	adantr 482	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑧 ∈ 𝐵)
120		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (comp‘𝐶) = (comp‘𝐶)
121	114	adantr 482	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → 𝑦 ∈ 𝐵)
122		simpr 486	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . 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 18215	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ) = (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔))
125	2, 31, 117, 119, 75, 118, 120, 121, 123, 122	yon2 18216	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → ((((𝑧(2^nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔) = (ℎ(⟨𝑦, 𝑧⟩(comp‘𝐶)𝑤)𝑔))
126	124, 125	eqtr4d 2776	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)) → (((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ) = ((((𝑧(2^nd ‘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(2^nd ‘((1^st ‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ)) = (𝑔 ∈ ((1^st ‘((1^st ‘𝑌)‘𝑧))‘𝑦) ↦ ((((𝑧(2^nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔)))
128	16	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌))
129	31, 75, 98, 128, 64, 62	funcf2 17815	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2^nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)⟶(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤)))
130	129	ffvelcdmda 7084	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2^nd ‘𝑌)𝑤)‘ℎ) ∈ (((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤)))
131	97, 130	nat1st2nd 17899	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2^nd ‘𝑌)𝑤)‘ℎ) ∈ (⟨(1^st ‘((1^st ‘𝑌)‘𝑧)), (2^nd ‘((1^st ‘𝑌)‘𝑧))⟩(𝑂 Nat 𝑆)⟨(1^st ‘((1^st ‘𝑌)‘𝑤)), (2^nd ‘((1^st ‘𝑌)‘𝑤))⟩))
132	131	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((𝑧(2^nd ‘𝑌)𝑤)‘ℎ) ∈ (⟨(1^st ‘((1^st ‘𝑌)‘𝑧)), (2^nd ‘((1^st ‘𝑌)‘𝑧))⟩(𝑂 Nat 𝑆)⟨(1^st ‘((1^st ‘𝑌)‘𝑤)), (2^nd ‘((1^st ‘𝑌)‘𝑤))⟩))
133		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
134	97, 132, 52, 133, 114	natcl 17901	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (((𝑧(2^nd ‘𝑌)𝑤)‘ℎ)‘𝑦) ∈ (((1^st ‘((1^st ‘𝑌)‘𝑧))‘𝑦)(Hom ‘𝑆)((1^st ‘((1^st ‘𝑌)‘𝑤))‘𝑦)))
135	10	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → 𝑈 ∈ V)
136	135	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 ∈ V)
137	60	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1^st ‘𝑌):𝐵⟶(𝑂 Func 𝑆))
138	137, 112	ffvelcdmd 7085	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1^st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆))
139		1st2ndbr 8025	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Rel (𝑂 Func 𝑆) ∧ ((1^st ‘𝑌)‘𝑧) ∈ (𝑂 Func 𝑆)) → (1^st ‘((1^st ‘𝑌)‘𝑧))(𝑂 Func 𝑆)(2^nd ‘((1^st ‘𝑌)‘𝑧)))
140	86, 138, 139	sylancr 588	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1^st ‘((1^st ‘𝑌)‘𝑧))(𝑂 Func 𝑆)(2^nd ‘((1^st ‘𝑌)‘𝑧)))
141	52, 85, 140	funcf1 17813	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1^st ‘((1^st ‘𝑌)‘𝑧)):𝐵⟶(Base‘𝑆))
142	141	ffvelcdmda 7084	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1^st ‘((1^st ‘𝑌)‘𝑧))‘𝑦) ∈ (Base‘𝑆))
143	92	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → 𝑈 = (Base‘𝑆))
144	142, 143	eleqtrrd 2837	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1^st ‘((1^st ‘𝑌)‘𝑧))‘𝑦) ∈ 𝑈)
145	89	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (1^st ‘((1^st ‘𝑌)‘𝑤)):𝐵⟶(Base‘𝑆))
146	145	ffvelcdmda 7084	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1^st ‘((1^st ‘𝑌)‘𝑤))‘𝑦) ∈ (Base‘𝑆))
147	146, 143	eleqtrrd 2837	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((1^st ‘((1^st ‘𝑌)‘𝑤))‘𝑦) ∈ 𝑈)
148	5, 136, 133, 144, 147	elsetchom 18028	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → ((((𝑧(2^nd ‘𝑌)𝑤)‘ℎ)‘𝑦) ∈ (((1^st ‘((1^st ‘𝑌)‘𝑧))‘𝑦)(Hom ‘𝑆)((1^st ‘((1^st ‘𝑌)‘𝑤))‘𝑦)) ↔ (((𝑧(2^nd ‘𝑌)𝑤)‘ℎ)‘𝑦):((1^st ‘((1^st ‘𝑌)‘𝑧))‘𝑦)⟶((1^st ‘((1^st ‘𝑌)‘𝑤))‘𝑦)))
149	134, 148	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (((𝑧(2^nd ‘𝑌)𝑤)‘ℎ)‘𝑦):((1^st ‘((1^st ‘𝑌)‘𝑧))‘𝑦)⟶((1^st ‘((1^st ‘𝑌)‘𝑤))‘𝑦))
150	149	feqmptd 6958	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (((𝑧(2^nd ‘𝑌)𝑤)‘ℎ)‘𝑦) = (𝑔 ∈ ((1^st ‘((1^st ‘𝑌)‘𝑧))‘𝑦) ↦ ((((𝑧(2^nd ‘𝑌)𝑤)‘ℎ)‘𝑦)‘𝑔)))
151	127, 150	eqtr4d 2776	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) ∧ 𝑦 ∈ 𝐵) → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ)) = (((𝑧(2^nd ‘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(2^nd ‘((1^st ‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ))) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2^nd ‘𝑌)𝑤)‘ℎ)‘𝑦)))
153	78	adantr 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → 𝑉 ∈ 𝑊)
154	79	adantr 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ran (Hom_f ‘𝐶) ⊆ 𝑈)
155	80	adantr 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → (ran (Hom_f ‘𝑄) ∪ 𝑈) ⊆ 𝑉)
156	63	adantr 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((1^st ‘𝑌)‘𝑤) ∈ (𝑂 Func 𝑆))
157	76	eleq2d 2820	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ℎ ∈ ((1^st ‘((1^st ‘𝑌)‘𝑤))‘𝑧) ↔ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)))
158	157	biimpar 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ℎ ∈ ((1^st ‘((1^st ‘𝑌)‘𝑤))‘𝑧))
159	2, 31, 32, 4, 5, 33, 6, 34, 28, 35, 36, 110, 153, 154, 155, 156, 112, 47, 158	yonedalem4a 18225	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1^st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧) ↦ (((𝑧(2^nd ‘((1^st ‘𝑌)‘𝑤))𝑦)‘𝑔)‘ℎ))))
160	97, 131, 52	natfn 17902	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2^nd ‘𝑌)𝑤)‘ℎ) Fn 𝐵)
161		dffn5 6948	. . . . . . . . . . 11 ⊢ (((𝑧(2^nd ‘𝑌)𝑤)‘ℎ) Fn 𝐵 ↔ ((𝑧(2^nd ‘𝑌)𝑤)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2^nd ‘𝑌)𝑤)‘ℎ)‘𝑦)))
162	160, 161	sylib 217	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((𝑧(2^nd ‘𝑌)𝑤)‘ℎ) = (𝑦 ∈ 𝐵 ↦ (((𝑧(2^nd ‘𝑌)𝑤)‘ℎ)‘𝑦)))
163	152, 159, 162	3eqtr4d 2783	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) ∧ ℎ ∈ (𝑧(Hom ‘𝐶)𝑤)) → ((((1^st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ) = ((𝑧(2^nd ‘𝑌)𝑤)‘ℎ))
164	163	mpteq2dva 5248	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1^st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ)) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2^nd ‘𝑌)𝑤)‘ℎ)))
165		f1of 6831	. . . . . . . . . 10 ⊢ ((((1^st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤)) → (((1^st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)⟶(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤)))
166	109, 165	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1^st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)⟶(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤)))
167	166	feqmptd 6958	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1^st ‘𝑌)‘𝑤)𝑁𝑧) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((((1^st ‘𝑌)‘𝑤)𝑁𝑧)‘ℎ)))
168	129	feqmptd 6958	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2^nd ‘𝑌)𝑤) = (ℎ ∈ (𝑧(Hom ‘𝐶)𝑤) ↦ ((𝑧(2^nd ‘𝑌)𝑤)‘ℎ)))
169	164, 167, 168	3eqtr4d 2783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((1^st ‘𝑌)‘𝑤)𝑁𝑧) = (𝑧(2^nd ‘𝑌)𝑤))
170	169	f1oeq1d 6826	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((((1^st ‘𝑌)‘𝑤)𝑁𝑧):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤)) ↔ (𝑧(2^nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤))))
171	109, 170	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(2^nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤)))
172	171	ralrimivva 3201	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧(2^nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤)))
173	31, 75, 98	isffth2 17864	. . . 4 ⊢ ((1^st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2^nd ‘𝑌) ↔ ((1^st ‘𝑌)(𝐶 Func 𝑄)(2^nd ‘𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧(2^nd ‘𝑌)𝑤):(𝑧(Hom ‘𝐶)𝑤)–1-1-onto→(((1^st ‘𝑌)‘𝑧)(𝑂 Nat 𝑆)((1^st ‘𝑌)‘𝑤))))
174	16, 172, 173	sylanbrc 584	. . 3 ⊢ (𝜑 → (1^st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2^nd ‘𝑌))
175		df-br 5149	. . 3 ⊢ ((1^st ‘𝑌)((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄))(2^nd ‘𝑌) ↔ ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
176	174, 175	sylib 217	. 2 ⊢ (𝜑 → ⟨(1^st ‘𝑌), (2^nd ‘𝑌)⟩ ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))
177	14, 176	eqeltrd 2834	1 ⊢ (𝜑 → 𝑌 ∈ ((𝐶 Full 𝑄) ∩ (𝐶 Faith 𝑄)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ⟨cop 4634 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ran crn 5677 Rel wrel 5681 Fn wfn 6536 ⟶wf 6537 –1-1-onto→wf1o 6540 ‘cfv 6541 (class class class)co 7406 ∈ cmpo 7408 1^st c1st 7970 2^nd c2nd 7971 tpos ctpos 8207 Basecbs 17141 Hom chom 17205 compcco 17206 Catccat 17605 Idccid 17606 Hom_f chomf 17607 oppCatcoppc 17652 Invcinv 17689 Isociso 17690 Func cfunc 17801 ∘_func ccofu 17803 Full cful 17850 Faith cfth 17851 Nat cnat 17889 FuncCat cfuc 17890 SetCatcsetc 18022 ×_c cxpc 18117 1^st_F c1stf 18118 2^nd_F c2ndf 18119 ⟨,⟩_F cprf 18120 eval_F cevlf 18159 Hom_Fchof 18198 Yoncyon 18199

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-hom 17218 df-cco 17219 df-cat 17609 df-cid 17610 df-homf 17611 df-comf 17612 df-oppc 17653 df-sect 17691 df-inv 17692 df-iso 17693 df-ssc 17754 df-resc 17755 df-subc 17756 df-func 17805 df-cofu 17807 df-full 17852 df-fth 17853 df-nat 17891 df-fuc 17892 df-setc 18023 df-xpc 18121 df-1stf 18122 df-2ndf 18123 df-prf 18124 df-evlf 18163 df-curf 18164 df-hof 18200 df-yon 18201

This theorem is referenced by: yonffth 18234

W3C validator