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Theorem yonedalem4c 18323
Description: Lemma for yoneda 18329. (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 18321 . . . 4 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))))
21 oveq1 7439 . . . . . 6 (𝑦 = 𝑧 → (𝑦(Hom ‘𝐶)𝑋) = (𝑧(Hom ‘𝐶)𝑋))
22 oveq2 7440 . . . . . . . 8 (𝑦 = 𝑧 → (𝑋(2nd𝐹)𝑦) = (𝑋(2nd𝐹)𝑧))
2322fveq1d 6907 . . . . . . 7 (𝑦 = 𝑧 → ((𝑋(2nd𝐹)𝑦)‘𝑔) = ((𝑋(2nd𝐹)𝑧)‘𝑔))
2423fveq1d 6907 . . . . . 6 (𝑦 = 𝑧 → (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴) = (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))
2521, 24mpteq12dv 5232 . . . . 5 (𝑦 = 𝑧 → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
2625cbvmptv 5254 . . . 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 2792 . . 3 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))))
284, 2oppcbas 17762 . . . . . . . . . . . . 13 𝐵 = (Base‘𝑂)
29 eqid 2736 . . . . . . . . . . . . 13 (Hom ‘𝑂) = (Hom ‘𝑂)
30 eqid 2736 . . . . . . . . . . . . 13 (Hom ‘𝑆) = (Hom ‘𝑆)
31 relfunc 17908 . . . . . . . . . . . . . . 15 Rel (𝑂 Func 𝑆)
32 1st2ndbr 8068 . . . . . . . . . . . . . . 15 ((Rel (𝑂 Func 𝑆) ∧ 𝐹 ∈ (𝑂 Func 𝑆)) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
3331, 16, 32sylancr 587 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
3433adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
3517adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝑋𝐵)
36 simpr 484 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝑧𝐵)
3728, 29, 30, 34, 35, 36funcf2 17914 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → (𝑋(2nd𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
3837adantr 480 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (𝑋(2nd𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
39 simpr 484 . . . . . . . . . . . 12 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋))
40 eqid 2736 . . . . . . . . . . . . 13 (Hom ‘𝐶) = (Hom ‘𝐶)
4140, 4oppchom 17759 . . . . . . . . . . . 12 (𝑋(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑋)
4239, 41eleqtrrdi 2851 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑔 ∈ (𝑋(Hom ‘𝑂)𝑧))
4338, 42ffvelcdmd 7104 . . . . . . . . . 10 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd𝐹)𝑧)‘𝑔) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
4415unssbd 4193 . . . . . . . . . . . . . 14 (𝜑𝑈𝑉)
4513, 44ssexd 5323 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ V)
4645adantr 480 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → 𝑈 ∈ V)
4746adantr 480 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑈 ∈ V)
48 eqid 2736 . . . . . . . . . . . . . . 15 (Base‘𝑆) = (Base‘𝑆)
4928, 48, 33funcf1 17912 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
505, 45setcbas 18124 . . . . . . . . . . . . . . 15 (𝜑𝑈 = (Base‘𝑆))
5150feq3d 6722 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹):𝐵𝑈 ↔ (1st𝐹):𝐵⟶(Base‘𝑆)))
5249, 51mpbird 257 . . . . . . . . . . . . 13 (𝜑 → (1st𝐹):𝐵𝑈)
5352, 17ffvelcdmd 7104 . . . . . . . . . . . 12 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
5453ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
5552ffvelcdmda 7103 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → ((1st𝐹)‘𝑧) ∈ 𝑈)
5655adantr 480 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑧) ∈ 𝑈)
575, 47, 30, 54, 56elsetchom 18127 . . . . . . . . . 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 232 . . . . . . . . 9 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd𝐹)𝑧)‘𝑔):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑧))
5919ad2antrr 726 . . . . . . . . 9 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐴 ∈ ((1st𝐹)‘𝑋))
6058, 59ffvelcdmd 7104 . . . . . . . 8 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴) ∈ ((1st𝐹)‘𝑧))
6160fmpttd 7134 . . . . . . 7 ((𝜑𝑧𝐵) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)):(𝑧(Hom ‘𝐶)𝑋)⟶((1st𝐹)‘𝑧))
6212adantr 480 . . . . . . . . 9 ((𝜑𝑧𝐵) → 𝐶 ∈ Cat)
631, 2, 62, 35, 40, 36yon11 18310 . . . . . . . 8 ((𝜑𝑧𝐵) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) = (𝑧(Hom ‘𝐶)𝑋))
6463feq2d 6721 . . . . . . 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 257 . . . . . 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 18309 . . . . . . . . . . 11 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
67 1st2ndbr 8068 . . . . . . . . . . 11 ((Rel (𝑂 Func 𝑆) ∧ ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆)) → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
6831, 66, 67sylancr 587 . . . . . . . . . 10 (𝜑 → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
6928, 48, 68funcf1 17912 . . . . . . . . 9 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
7050feq3d 6722 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑋)):𝐵𝑈 ↔ (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆)))
7169, 70mpbird 257 . . . . . . . 8 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵𝑈)
7271ffvelcdmda 7103 . . . . . . 7 ((𝜑𝑧𝐵) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ∈ 𝑈)
735, 46, 30, 72, 55elsetchom 18127 . . . . . 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 257 . . . . 5 ((𝜑𝑧𝐵) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
7574ralrimiva 3145 . . . 4 (𝜑 → ∀𝑧𝐵 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
762fvexi 6919 . . . . 5 𝐵 ∈ V
77 mptelixpg 8976 . . . . 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 234 . . 3 (𝜑 → (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))) ∈ X𝑧𝐵 (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
8027, 79eqeltrd 2840 . 2 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ X𝑧𝐵 (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
8112adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝐶 ∈ Cat)
8217adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑋𝐵)
83 simpr1 1194 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑧𝐵)
841, 2, 81, 82, 40, 83yon11 18310 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) = (𝑧(Hom ‘𝐶)𝑋))
8584eleq2d 2826 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ↔ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)))
8685biimpa 476 . . . . . . 7 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑧)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋))
87 eqid 2736 . . . . . . . . . . . 12 (comp‘𝑂) = (comp‘𝑂)
88 eqid 2736 . . . . . . . . . . . 12 (comp‘𝑆) = (comp‘𝑆)
8933adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
9089adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
9182adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑋𝐵)
9283adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑧𝐵)
93 simpr2 1195 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑤𝐵)
9493adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑤𝐵)
95 simpr 484 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋))
9695, 41eleqtrrdi 2851 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑘 ∈ (𝑋(Hom ‘𝑂)𝑧))
97 simplr3 1217 . . . . . . . . . . . 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 17917 . . . . . . . . . . 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 2736 . . . . . . . . . . . . 13 (comp‘𝐶) = (comp‘𝐶)
1002, 99, 4, 91, 92, 94oppcco 17761 . . . . . . . . . . . 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 6909 . . . . . . . . . . 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 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑈 ∈ V)
103102adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑈 ∈ V)
10453ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
105553ad2antr1 1188 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st𝐹)‘𝑧) ∈ 𝑈)
106105adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑧) ∈ 𝑈)
10752adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st𝐹):𝐵𝑈)
108107, 93ffvelcdmd 7104 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st𝐹)‘𝑤) ∈ 𝑈)
109108adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑤) ∈ 𝑈)
11028, 29, 30, 89, 82, 83funcf2 17914 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑋(2nd𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
111110adantr 480 . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . 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 18127 . . . . . . . . . . . . 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 232 . . . . . . . . . . . 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 17914 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑧(2nd𝐹)𝑤):(𝑧(Hom ‘𝑂)𝑤)⟶(((1st𝐹)‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑤)))
116 simpr3 1196 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ∈ (𝑧(Hom ‘𝑂)𝑤))
117115, 116ffvelcdmd 7104 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd𝐹)𝑤)‘) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑤)))
1185, 102, 30, 105, 108elsetchom 18127 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd𝐹)𝑤)‘):((1st𝐹)‘𝑧)⟶((1st𝐹)‘𝑤))
120119adantr 480 . . . . . . . . . . . 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 18129 . . . . . . . . . . 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 2784 . . . . . . . . . 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 6907 . . . . . . . . 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 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐴 ∈ ((1st𝐹)‘𝑋))
125 fvco3 7007 . . . . . . . . . 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 584 . . . . . . . . 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 2776 . . . . . . . 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 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐶 ∈ Cat)
12940, 4oppchom 17759 . . . . . . . . . . . 12 (𝑧(Hom ‘𝑂)𝑤) = (𝑤(Hom ‘𝐶)𝑧)
13097, 129eleqtrdi 2850 . . . . . . . . . . 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 18311 . . . . . . . . . 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 6909 . . . . . . . . 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 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑉𝑊)
13414ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ran (Homf𝐶) ⊆ 𝑈)
13515ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
13616ad2antrr 726 . . . . . . . . . 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 17729 . . . . . . . . . 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 18322 . . . . . . . . 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 2776 . . . . . . . 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 18322 . . . . . . . . 9 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘) = (((𝑋(2nd𝐹)𝑧)‘𝑘)‘𝐴))
141140fveq2d 6909 . . . . . . . 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 2786 . . . . . . 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 591 . . . . . 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 5241 . . . . 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 6905 . . . . . . . 8 (𝑧 = 𝑤 → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = (((𝐹𝑁𝑋)‘𝐴)‘𝑤))
146 fveq2 6905 . . . . . . . 8 (𝑧 = 𝑤 → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) = ((1st ‘((1st𝑌)‘𝑋))‘𝑤))
147 fveq2 6905 . . . . . . . 8 (𝑧 = 𝑤 → ((1st𝐹)‘𝑧) = ((1st𝐹)‘𝑤))
148145, 146, 147feq123d 6724 . . . . . . 7 (𝑧 = 𝑤 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧) ↔ (((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1st ‘((1st𝑌)‘𝑋))‘𝑤)⟶((1st𝐹)‘𝑤)))
14927fveq1d 6907 . . . . . . . . . . . 12 (𝜑 → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = ((𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))‘𝑧))
150 ovex 7465 . . . . . . . . . . . . . 14 (𝑧(Hom ‘𝐶)𝑋) ∈ V
151150mptex 7244 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ V
152 eqid 2736 . . . . . . . . . . . . . 14 (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))) = (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
153152fvmpt2 7026 . . . . . . . . . . . . 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 691 . . . . . . . . . . . 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 2796 . . . . . . . . . . 11 ((𝜑𝑧𝐵) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
156155feq1d 6719 . . . . . . . . . 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 257 . . . . . . . . 9 ((𝜑𝑧𝐵) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
158157ralrimiva 3145 . . . . . . . 8 (𝜑 → ∀𝑧𝐵 (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
159158adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ∀𝑧𝐵 (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
160148, 159, 93rspcdva 3622 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1st ‘((1st𝑌)‘𝑋))‘𝑤)⟶((1st𝐹)‘𝑤))
16168adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
16228, 29, 30, 161, 83, 93funcf2 17914 . . . . . . . 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 7104 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑤)))
164723ad2antr1 1188 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ∈ 𝑈)
16571adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st ‘((1st𝑌)‘𝑋)):𝐵𝑈)
166165, 93ffvelcdmd 7104 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st ‘((1st𝑌)‘𝑋))‘𝑤) ∈ 𝑈)
1675, 102, 30, 164, 166elsetchom 18127 . . . . . . 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 232 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑤))
169 fcompt 7152 . . . . . 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 584 . . . . 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 1188 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
172 fcompt 7152 . . . . . 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 584 . . . . 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 2786 . . . 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 18129 . . . 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 18129 . . . 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 2786 . . 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 3204 . 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 2736 . . 3 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
180179, 28, 29, 30, 88, 66, 16isnat2 17997 . 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 713 1 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  cun 3948  wss 3950  cop 4631   class class class wbr 5142  cmpt 5224  ran crn 5685  ccom 5688  Rel wrel 5689  wf 6556  cfv 6560  (class class class)co 7432  cmpo 7434  1st c1st 8013  2nd c2nd 8014  tpos ctpos 8251  Xcixp 8938  Basecbs 17248  Hom chom 17309  compcco 17310  Catccat 17708  Idccid 17709  Homf chomf 17710  oppCatcoppc 17755   Func cfunc 17900  func ccofu 17902   Nat cnat 17990   FuncCat cfuc 17991  SetCatcsetc 18121   ×c cxpc 18214   1stF c1stf 18215   2ndF c2ndf 18216   ⟨,⟩F cprf 18217   evalF cevlf 18255  HomFchof 18294  Yoncyon 18295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-hom 17322  df-cco 17323  df-cat 17712  df-cid 17713  df-homf 17714  df-comf 17715  df-oppc 17756  df-func 17904  df-nat 17992  df-fuc 17993  df-setc 18122  df-xpc 18218  df-curf 18260  df-hof 18296  df-yon 18297
This theorem is referenced by:  yonedainv  18327
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