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Theorem yonedalem4c 18302
Description: Lemma for yoneda 18308. (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 = (HomF𝑄)
yoneda.r 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
yoneda.e 𝐸 = (𝑂 evalF 𝑆)
yoneda.z 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
yoneda.c (𝜑𝐶 ∈ Cat)
yoneda.w (𝜑𝑉𝑊)
yoneda.u (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
yoneda.v (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
yonedalem21.f (𝜑𝐹 ∈ (𝑂 Func 𝑆))
yonedalem21.x (𝜑𝑋𝐵)
yonedalem4.n 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
yonedalem4.p (𝜑𝐴 ∈ ((1st𝐹)‘𝑋))
Assertion
Ref Expression
yonedalem4c (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦, 1   𝑢,𝑔,𝐴,𝑦   𝑢,𝑓,𝐶,𝑔,𝑥,𝑦   𝑓,𝐸,𝑔,𝑢,𝑦   𝑓,𝐹,𝑔,𝑢,𝑥,𝑦   𝐵,𝑓,𝑔,𝑢,𝑥,𝑦   𝑓,𝑂,𝑔,𝑢,𝑥,𝑦   𝑆,𝑓,𝑔,𝑢,𝑥,𝑦   𝑄,𝑓,𝑔,𝑢,𝑥   𝑇,𝑓,𝑔,𝑢,𝑦   𝜑,𝑓,𝑔,𝑢,𝑥,𝑦   𝑢,𝑅   𝑓,𝑌,𝑔,𝑢,𝑥,𝑦   𝑓,𝑍,𝑔,𝑢,𝑥,𝑦   𝑓,𝑋,𝑔,𝑢,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑓)   𝑄(𝑦)   𝑅(𝑥,𝑦,𝑓,𝑔)   𝑇(𝑥)   𝑈(𝑥,𝑦,𝑢,𝑓,𝑔)   1 (𝑢)   𝐸(𝑥)   𝐻(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑁(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑊(𝑥,𝑦,𝑢,𝑓,𝑔)

Proof of Theorem yonedalem4c
Dummy variables 𝑘 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 yoneda.y . . . . 5 𝑌 = (Yon‘𝐶)
2 yoneda.b . . . . 5 𝐵 = (Base‘𝐶)
3 yoneda.1 . . . . 5 1 = (Id‘𝐶)
4 yoneda.o . . . . 5 𝑂 = (oppCat‘𝐶)
5 yoneda.s . . . . 5 𝑆 = (SetCat‘𝑈)
6 yoneda.t . . . . 5 𝑇 = (SetCat‘𝑉)
7 yoneda.q . . . . 5 𝑄 = (𝑂 FuncCat 𝑆)
8 yoneda.h . . . . 5 𝐻 = (HomF𝑄)
9 yoneda.r . . . . 5 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇)
10 yoneda.e . . . . 5 𝐸 = (𝑂 evalF 𝑆)
11 yoneda.z . . . . 5 𝑍 = (𝐻func ((⟨(1st𝑌), tpos (2nd𝑌)⟩ ∘func (𝑄 2ndF 𝑂)) ⟨,⟩F (𝑄 1stF 𝑂)))
12 yoneda.c . . . . 5 (𝜑𝐶 ∈ Cat)
13 yoneda.w . . . . 5 (𝜑𝑉𝑊)
14 yoneda.u . . . . 5 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
15 yoneda.v . . . . 5 (𝜑 → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
16 yonedalem21.f . . . . 5 (𝜑𝐹 ∈ (𝑂 Func 𝑆))
17 yonedalem21.x . . . . 5 (𝜑𝑋𝐵)
18 yonedalem4.n . . . . 5 𝑁 = (𝑓 ∈ (𝑂 Func 𝑆), 𝑥𝐵 ↦ (𝑢 ∈ ((1st𝑓)‘𝑥) ↦ (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥) ↦ (((𝑥(2nd𝑓)𝑦)‘𝑔)‘𝑢)))))
19 yonedalem4.p . . . . 5 (𝜑𝐴 ∈ ((1st𝐹)‘𝑋))
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19yonedalem4a 18300 . . . 4 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))))
21 oveq1 7431 . . . . . 6 (𝑦 = 𝑧 → (𝑦(Hom ‘𝐶)𝑋) = (𝑧(Hom ‘𝐶)𝑋))
22 oveq2 7432 . . . . . . . 8 (𝑦 = 𝑧 → (𝑋(2nd𝐹)𝑦) = (𝑋(2nd𝐹)𝑧))
2322fveq1d 6903 . . . . . . 7 (𝑦 = 𝑧 → ((𝑋(2nd𝐹)𝑦)‘𝑔) = ((𝑋(2nd𝐹)𝑧)‘𝑔))
2423fveq1d 6903 . . . . . 6 (𝑦 = 𝑧 → (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴) = (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))
2521, 24mpteq12dv 5244 . . . . 5 (𝑦 = 𝑧 → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
2625cbvmptv 5266 . . . 4 (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))) = (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
2720, 26eqtrdi 2782 . . 3 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))))
284, 2oppcbas 17732 . . . . . . . . . . . . 13 𝐵 = (Base‘𝑂)
29 eqid 2726 . . . . . . . . . . . . 13 (Hom ‘𝑂) = (Hom ‘𝑂)
30 eqid 2726 . . . . . . . . . . . . 13 (Hom ‘𝑆) = (Hom ‘𝑆)
31 relfunc 17881 . . . . . . . . . . . . . . 15 Rel (𝑂 Func 𝑆)
32 1st2ndbr 8056 . . . . . . . . . . . . . . 15 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
3331, 16, 32sylancr 585 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
3433adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
3517adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝑋𝐵)
36 simpr 483 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝑧𝐵)
3728, 29, 30, 34, 35, 36funcf2 17887 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → (𝑋(2nd𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
3837adantr 479 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (𝑋(2nd𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
39 simpr 483 . . . . . . . . . . . 12 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋))
40 eqid 2726 . . . . . . . . . . . . 13 (Hom ‘𝐶) = (Hom ‘𝐶)
4140, 4oppchom 17729 . . . . . . . . . . . 12 (𝑋(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑋)
4239, 41eleqtrrdi 2837 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑔 ∈ (𝑋(Hom ‘𝑂)𝑧))
4338, 42ffvelcdmd 7099 . . . . . . . . . 10 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd𝐹)𝑧)‘𝑔) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
4415unssbd 4189 . . . . . . . . . . . . . 14 (𝜑𝑈𝑉)
4513, 44ssexd 5329 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ V)
4645adantr 479 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → 𝑈 ∈ V)
4746adantr 479 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑈 ∈ V)
48 eqid 2726 . . . . . . . . . . . . . . 15 (Base‘𝑆) = (Base‘𝑆)
4928, 48, 33funcf1 17885 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
505, 45setcbas 18100 . . . . . . . . . . . . . . 15 (𝜑𝑈 = (Base‘𝑆))
5150feq3d 6715 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹):𝐵𝑈 ↔ (1st𝐹):𝐵⟶(Base‘𝑆)))
5249, 51mpbird 256 . . . . . . . . . . . . 13 (𝜑 → (1st𝐹):𝐵𝑈)
5352, 17ffvelcdmd 7099 . . . . . . . . . . . 12 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
5453ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
5552ffvelcdmda 7098 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → ((1st𝐹)‘𝑧) ∈ 𝑈)
5655adantr 479 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑧) ∈ 𝑈)
575, 47, 30, 54, 56elsetchom 18103 . . . . . . . . . 10 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2nd𝐹)𝑧)‘𝑔) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)) ↔ ((𝑋(2nd𝐹)𝑧)‘𝑔):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑧)))
5843, 57mpbid 231 . . . . . . . . 9 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd𝐹)𝑧)‘𝑔):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑧))
5919ad2antrr 724 . . . . . . . . 9 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐴 ∈ ((1st𝐹)‘𝑋))
6058, 59ffvelcdmd 7099 . . . . . . . 8 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴) ∈ ((1st𝐹)‘𝑧))
6160fmpttd 7129 . . . . . . 7 ((𝜑𝑧𝐵) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)):(𝑧(Hom ‘𝐶)𝑋)⟶((1st𝐹)‘𝑧))
6212adantr 479 . . . . . . . . 9 ((𝜑𝑧𝐵) → 𝐶 ∈ Cat)
631, 2, 62, 35, 40, 36yon11 18289 . . . . . . . 8 ((𝜑𝑧𝐵) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) = (𝑧(Hom ‘𝐶)𝑋))
6463feq2d 6714 . . . . . . 7 ((𝜑𝑧𝐵) → ((𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧) ↔ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)):(𝑧(Hom ‘𝐶)𝑋)⟶((1st𝐹)‘𝑧)))
6561, 64mpbird 256 . . . . . 6 ((𝜑𝑧𝐵) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
661, 2, 12, 17, 4, 5, 45, 14yon1cl 18288 . . . . . . . . . . 11 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
67 1st2ndbr 8056 . . . . . . . . . . 11 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
6831, 66, 67sylancr 585 . . . . . . . . . 10 (𝜑 → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
6928, 48, 68funcf1 17885 . . . . . . . . 9 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
7050feq3d 6715 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑋)):𝐵𝑈 ↔ (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆)))
7169, 70mpbird 256 . . . . . . . 8 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵𝑈)
7271ffvelcdmda 7098 . . . . . . 7 ((𝜑𝑧𝐵) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ∈ 𝑈)
735, 46, 30, 72, 55elsetchom 18103 . . . . . 6 ((𝜑𝑧𝐵) → ((𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)) ↔ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧)))
7465, 73mpbird 256 . . . . 5 ((𝜑𝑧𝐵) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
7574ralrimiva 3136 . . . 4 (𝜑 → ∀𝑧𝐵 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
762fvexi 6915 . . . . 5 𝐵 ∈ V
77 mptelixpg 8964 . . . . 5 (𝐵 ∈ V → ((𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))) ∈ X𝑧𝐵 (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)) ↔ ∀𝑧𝐵 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧))))
7876, 77ax-mp 5 . . . 4 ((𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))) ∈ X𝑧𝐵 (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)) ↔ ∀𝑧𝐵 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
7975, 78sylibr 233 . . 3 (𝜑 → (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))) ∈ X𝑧𝐵 (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
8027, 79eqeltrd 2826 . 2 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ X𝑧𝐵 (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
8112adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝐶 ∈ Cat)
8217adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑋𝐵)
83 simpr1 1191 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑧𝐵)
841, 2, 81, 82, 40, 83yon11 18289 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) = (𝑧(Hom ‘𝐶)𝑋))
8584eleq2d 2812 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ↔ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)))
8685biimpa 475 . . . . . . 7 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑧)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋))
87 eqid 2726 . . . . . . . . . . . 12 (comp‘𝑂) = (comp‘𝑂)
88 eqid 2726 . . . . . . . . . . . 12 (comp‘𝑆) = (comp‘𝑆)
8933adantr 479 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
9089adantr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
9182adantr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑋𝐵)
9283adantr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑧𝐵)
93 simpr2 1192 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑤𝐵)
9493adantr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑤𝐵)
95 simpr 483 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋))
9695, 41eleqtrrdi 2837 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑘 ∈ (𝑋(Hom ‘𝑂)𝑧))
97 simplr3 1214 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ∈ (𝑧(Hom ‘𝑂)𝑤))
9828, 29, 87, 88, 90, 91, 92, 94, 96, 97funcco 17890 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd𝐹)𝑤)‘((⟨𝑋, 𝑧⟩(comp‘𝑂)𝑤)𝑘)) = (((𝑧(2nd𝐹)𝑤)‘)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑧)⟩(comp‘𝑆)((1st𝐹)‘𝑤))((𝑋(2nd𝐹)𝑧)‘𝑘)))
99 eqid 2726 . . . . . . . . . . . . 13 (comp‘𝐶) = (comp‘𝐶)
1002, 99, 4, 91, 92, 94oppcco 17731 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((⟨𝑋, 𝑧⟩(comp‘𝑂)𝑤)𝑘) = (𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)))
101100fveq2d 6905 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd𝐹)𝑤)‘((⟨𝑋, 𝑧⟩(comp‘𝑂)𝑤)𝑘)) = ((𝑋(2nd𝐹)𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋))))
10245adantr 479 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑈 ∈ V)
103102adantr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑈 ∈ V)
10453ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
105553ad2antr1 1185 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st𝐹)‘𝑧) ∈ 𝑈)
106105adantr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑧) ∈ 𝑈)
10752adantr 479 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st𝐹):𝐵𝑈)
108107, 93ffvelcdmd 7099 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st𝐹)‘𝑤) ∈ 𝑈)
109108adantr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑤) ∈ 𝑈)
11028, 29, 30, 89, 82, 83funcf2 17887 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑋(2nd𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
111110adantr 479 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (𝑋(2nd𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
112111, 96ffvelcdmd 7099 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd𝐹)𝑧)‘𝑘) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
1135, 103, 30, 104, 106elsetchom 18103 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2nd𝐹)𝑧)‘𝑘) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)) ↔ ((𝑋(2nd𝐹)𝑧)‘𝑘):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑧)))
114112, 113mpbid 231 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd𝐹)𝑧)‘𝑘):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑧))
11528, 29, 30, 89, 83, 93funcf2 17887 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑧(2nd𝐹)𝑤):(𝑧(Hom ‘𝑂)𝑤)⟶(((1st𝐹)‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑤)))
116 simpr3 1193 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ∈ (𝑧(Hom ‘𝑂)𝑤))
117115, 116ffvelcdmd 7099 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd𝐹)𝑤)‘) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑤)))
1185, 102, 30, 105, 108elsetchom 18103 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝑧(2nd𝐹)𝑤)‘) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑤)) ↔ ((𝑧(2nd𝐹)𝑤)‘):((1st𝐹)‘𝑧)⟶((1st𝐹)‘𝑤)))
119117, 118mpbid 231 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd𝐹)𝑤)‘):((1st𝐹)‘𝑧)⟶((1st𝐹)‘𝑤))
120119adantr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑧(2nd𝐹)𝑤)‘):((1st𝐹)‘𝑧)⟶((1st𝐹)‘𝑤))
1215, 103, 88, 104, 106, 109, 114, 120setcco 18105 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑧(2nd𝐹)𝑤)‘)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑧)⟩(comp‘𝑆)((1st𝐹)‘𝑤))((𝑋(2nd𝐹)𝑧)‘𝑘)) = (((𝑧(2nd𝐹)𝑤)‘) ∘ ((𝑋(2nd𝐹)𝑧)‘𝑘)))
12298, 101, 1213eqtr3d 2774 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd𝐹)𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋))) = (((𝑧(2nd𝐹)𝑤)‘) ∘ ((𝑋(2nd𝐹)𝑧)‘𝑘)))
123122fveq1d 6903 . . . . . . . . 9 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2nd𝐹)𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)))‘𝐴) = ((((𝑧(2nd𝐹)𝑤)‘) ∘ ((𝑋(2nd𝐹)𝑧)‘𝑘))‘𝐴))
12419ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐴 ∈ ((1st𝐹)‘𝑋))
125 fvco3 7001 . . . . . . . . . 10 ((((𝑋(2nd𝐹)𝑧)‘𝑘):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑧) ∧ 𝐴 ∈ ((1st𝐹)‘𝑋)) → ((((𝑧(2nd𝐹)𝑤)‘) ∘ ((𝑋(2nd𝐹)𝑧)‘𝑘))‘𝐴) = (((𝑧(2nd𝐹)𝑤)‘)‘(((𝑋(2nd𝐹)𝑧)‘𝑘)‘𝐴)))
126114, 124, 125syl2anc 582 . . . . . . . . 9 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝑧(2nd𝐹)𝑤)‘) ∘ ((𝑋(2nd𝐹)𝑧)‘𝑘))‘𝐴) = (((𝑧(2nd𝐹)𝑤)‘)‘(((𝑋(2nd𝐹)𝑧)‘𝑘)‘𝐴)))
127123, 126eqtrd 2766 . . . . . . . 8 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2nd𝐹)𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)))‘𝐴) = (((𝑧(2nd𝐹)𝑤)‘)‘(((𝑋(2nd𝐹)𝑧)‘𝑘)‘𝐴)))
12881adantr 479 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐶 ∈ Cat)
12940, 4oppchom 17729 . . . . . . . . . . . 12 (𝑧(Hom ‘𝑂)𝑤) = (𝑤(Hom ‘𝐶)𝑧)
13097, 129eleqtrdi 2836 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ∈ (𝑤(Hom ‘𝐶)𝑧))
1311, 2, 128, 91, 40, 92, 99, 94, 130, 95yon12 18290 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)‘𝑘) = (𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)))
132131fveq2d 6905 . . . . . . . . 9 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)‘𝑘)) = ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋))))
13313ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑉𝑊)
13414ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ran (Homf𝐶) ⊆ 𝑈)
13515ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
13616ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐹 ∈ (𝑂 Func 𝑆))
1372, 40, 99, 128, 94, 92, 91, 130, 95catcocl 17698 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)) ∈ (𝑤(Hom ‘𝐶)𝑋))
1381, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 128, 133, 134, 135, 136, 91, 18, 124, 94, 137yonedalem4b 18301 . . . . . . . . 9 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋))) = (((𝑋(2nd𝐹)𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)))‘𝐴))
139132, 138eqtrd 2766 . . . . . . . 8 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)‘𝑘)) = (((𝑋(2nd𝐹)𝑤)‘(𝑘(⟨𝑤, 𝑧⟩(comp‘𝐶)𝑋)))‘𝐴))
1401, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 128, 133, 134, 135, 136, 91, 18, 124, 92, 95yonedalem4b 18301 . . . . . . . . 9 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘) = (((𝑋(2nd𝐹)𝑧)‘𝑘)‘𝐴))
141140fveq2d 6905 . . . . . . . 8 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑧(2nd𝐹)𝑤)‘)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘)) = (((𝑧(2nd𝐹)𝑤)‘)‘(((𝑋(2nd𝐹)𝑧)‘𝑘)‘𝐴)))
142127, 139, 1413eqtr4d 2776 . . . . . . 7 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)‘𝑘)) = (((𝑧(2nd𝐹)𝑤)‘)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘)))
14386, 142syldan 589 . . . . . 6 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑧)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)‘𝑘)) = (((𝑧(2nd𝐹)𝑤)‘)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘)))
144143mpteq2dva 5253 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ↦ ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)‘𝑘))) = (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ↦ (((𝑧(2nd𝐹)𝑤)‘)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘))))
145 fveq2 6901 . . . . . . . 8 (𝑧 = 𝑤 → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = (((𝐹𝑁𝑋)‘𝐴)‘𝑤))
146 fveq2 6901 . . . . . . . 8 (𝑧 = 𝑤 → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) = ((1st ‘((1st𝑌)‘𝑋))‘𝑤))
147 fveq2 6901 . . . . . . . 8 (𝑧 = 𝑤 → ((1st𝐹)‘𝑧) = ((1st𝐹)‘𝑤))
148145, 146, 147feq123d 6717 . . . . . . 7 (𝑧 = 𝑤 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧) ↔ (((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1st ‘((1st𝑌)‘𝑋))‘𝑤)⟶((1st𝐹)‘𝑤)))
14927fveq1d 6903 . . . . . . . . . . . 12 (𝜑 → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = ((𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))‘𝑧))
150 ovex 7457 . . . . . . . . . . . . . 14 (𝑧(Hom ‘𝐶)𝑋) ∈ V
151150mptex 7240 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ V
152 eqid 2726 . . . . . . . . . . . . . 14 (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))) = (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
153152fvmpt2 7020 . . . . . . . . . . . . 13 ((𝑧𝐵 ∧ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ V) → ((𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))‘𝑧) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
154151, 153mpan2 689 . . . . . . . . . . . 12 (𝑧𝐵 → ((𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))‘𝑧) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
155149, 154sylan9eq 2786 . . . . . . . . . . 11 ((𝜑𝑧𝐵) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
156155feq1d 6713 . . . . . . . . . 10 ((𝜑𝑧𝐵) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧) ↔ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧)))
15765, 156mpbird 256 . . . . . . . . 9 ((𝜑𝑧𝐵) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
158157ralrimiva 3136 . . . . . . . 8 (𝜑 → ∀𝑧𝐵 (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
159158adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ∀𝑧𝐵 (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
160148, 159, 93rspcdva 3609 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1st ‘((1st𝑌)‘𝑋))‘𝑤)⟶((1st𝐹)‘𝑤))
16168adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
16228, 29, 30, 161, 83, 93funcf2 17887 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤):(𝑧(Hom ‘𝑂)𝑤)⟶(((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑤)))
163162, 116ffvelcdmd 7099 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑤)))
164723ad2antr1 1185 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ∈ 𝑈)
16571adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st ‘((1st𝑌)‘𝑋)):𝐵𝑈)
166165, 93ffvelcdmd 7099 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st ‘((1st𝑌)‘𝑋))‘𝑤) ∈ 𝑈)
1675, 102, 30, 164, 166elsetchom 18103 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑤)) ↔ ((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑤)))
168163, 167mpbid 231 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑤))
169 fcompt 7147 . . . . . 6 (((((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1st ‘((1st𝑌)‘𝑋))‘𝑤)⟶((1st𝐹)‘𝑤) ∧ ((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑤)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤) ∘ ((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)) = (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ↦ ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)‘𝑘))))
170160, 168, 169syl2anc 582 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤) ∘ ((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)) = (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ↦ ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)‘(((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)‘𝑘))))
1711573ad2antr1 1185 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
172 fcompt 7147 . . . . . 6 ((((𝑧(2nd𝐹)𝑤)‘):((1st𝐹)‘𝑧)⟶((1st𝐹)‘𝑤) ∧ (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧)) → (((𝑧(2nd𝐹)𝑤)‘) ∘ (((𝐹𝑁𝑋)‘𝐴)‘𝑧)) = (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ↦ (((𝑧(2nd𝐹)𝑤)‘)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘))))
173119, 171, 172syl2anc 582 . . . . 5 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝑧(2nd𝐹)𝑤)‘) ∘ (((𝐹𝑁𝑋)‘𝐴)‘𝑧)) = (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ↦ (((𝑧(2nd𝐹)𝑤)‘)‘((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘))))
174144, 170, 1733eqtr4d 2776 . . . 4 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤) ∘ ((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)) = (((𝑧(2nd𝐹)𝑤)‘) ∘ (((𝐹𝑁𝑋)‘𝐴)‘𝑧)))
1755, 102, 88, 164, 166, 108, 168, 160setcco 18105 . . . 4 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑧), ((1st ‘((1st𝑌)‘𝑋))‘𝑤)⟩(comp‘𝑆)((1st𝐹)‘𝑤))((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)) = ((((𝐹𝑁𝑋)‘𝐴)‘𝑤) ∘ ((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)))
1765, 102, 88, 164, 105, 108, 171, 119setcco 18105 . . . 4 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝑧(2nd𝐹)𝑤)‘)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑧), ((1st𝐹)‘𝑧)⟩(comp‘𝑆)((1st𝐹)‘𝑤))(((𝐹𝑁𝑋)‘𝐴)‘𝑧)) = (((𝑧(2nd𝐹)𝑤)‘) ∘ (((𝐹𝑁𝑋)‘𝐴)‘𝑧)))
177174, 175, 1763eqtr4d 2776 . . 3 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑤)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑧), ((1st ‘((1st𝑌)‘𝑋))‘𝑤)⟩(comp‘𝑆)((1st𝐹)‘𝑤))((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)) = (((𝑧(2nd𝐹)𝑤)‘)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑧), ((1st𝐹)‘𝑧)⟩(comp‘𝑆)((1st𝐹)‘𝑤))(((𝐹𝑁𝑋)‘𝐴)‘𝑧)))
178177ralrimivvva 3194 . 2 (𝜑 → ∀𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤)((((𝐹𝑁𝑋)‘𝐴)‘𝑤)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑧), ((1st ‘((1st𝑌)‘𝑋))‘𝑤)⟩(comp‘𝑆)((1st𝐹)‘𝑤))((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)) = (((𝑧(2nd𝐹)𝑤)‘)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑧), ((1st𝐹)‘𝑧)⟩(comp‘𝑆)((1st𝐹)‘𝑤))(((𝐹𝑁𝑋)‘𝐴)‘𝑧)))
179 eqid 2726 . . 3 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
180179, 28, 29, 30, 88, 66, 16isnat2 17971 . 2 (𝜑 → (((𝐹𝑁𝑋)‘𝐴) ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹) ↔ (((𝐹𝑁𝑋)‘𝐴) ∈ X𝑧𝐵 (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)) ∧ ∀𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤)((((𝐹𝑁𝑋)‘𝐴)‘𝑤)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑧), ((1st ‘((1st𝑌)‘𝑋))‘𝑤)⟩(comp‘𝑆)((1st𝐹)‘𝑤))((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘)) = (((𝑧(2nd𝐹)𝑤)‘)(⟨((1st ‘((1st𝑌)‘𝑋))‘𝑧), ((1st𝐹)‘𝑧)⟩(comp‘𝑆)((1st𝐹)‘𝑤))(((𝐹𝑁𝑋)‘𝐴)‘𝑧)))))
18180, 178, 180mpbir2and 711 1 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  Vcvv 3462  cun 3945  wss 3947  cop 4639   class class class wbr 5153  cmpt 5236  ran crn 5683  ccom 5686  Rel wrel 5687  wf 6550  cfv 6554  (class class class)co 7424  cmpo 7426  1st c1st 8001  2nd c2nd 8002  tpos ctpos 8240  Xcixp 8926  Basecbs 17213  Hom chom 17277  compcco 17278  Catccat 17677  Idccid 17678  Homf chomf 17679  oppCatcoppc 17724   Func cfunc 17873  func ccofu 17875   Nat cnat 17964   FuncCat cfuc 17965  SetCatcsetc 18097   ×c cxpc 18192   1stF c1stf 18193   2ndF c2ndf 18194   ⟨,⟩F cprf 18195   evalF cevlf 18234  HomFchof 18273  Yoncyon 18274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-tpos 8241  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-map 8857  df-ixp 8927  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  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 12611  df-dec 12730  df-uz 12875  df-fz 13539  df-struct 17149  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-hom 17290  df-cco 17291  df-cat 17681  df-cid 17682  df-homf 17683  df-comf 17684  df-oppc 17725  df-func 17877  df-nat 17966  df-fuc 17967  df-setc 18098  df-xpc 18196  df-curf 18239  df-hof 18275  df-yon 18276
This theorem is referenced by:  yonedainv  18306
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