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Theorem yonedalem4c 18334
Description: Lemma for yoneda 18340. (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 18332 . . . 4 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))))
21 oveq1 7438 . . . . . 6 (𝑦 = 𝑧 → (𝑦(Hom ‘𝐶)𝑋) = (𝑧(Hom ‘𝐶)𝑋))
22 oveq2 7439 . . . . . . . 8 (𝑦 = 𝑧 → (𝑋(2nd𝐹)𝑦) = (𝑋(2nd𝐹)𝑧))
2322fveq1d 6909 . . . . . . 7 (𝑦 = 𝑧 → ((𝑋(2nd𝐹)𝑦)‘𝑔) = ((𝑋(2nd𝐹)𝑧)‘𝑔))
2423fveq1d 6909 . . . . . 6 (𝑦 = 𝑧 → (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴) = (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))
2521, 24mpteq12dv 5239 . . . . 5 (𝑦 = 𝑧 → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
2625cbvmptv 5261 . . . 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 2791 . . 3 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))))
284, 2oppcbas 17764 . . . . . . . . . . . . 13 𝐵 = (Base‘𝑂)
29 eqid 2735 . . . . . . . . . . . . 13 (Hom ‘𝑂) = (Hom ‘𝑂)
30 eqid 2735 . . . . . . . . . . . . 13 (Hom ‘𝑆) = (Hom ‘𝑆)
31 relfunc 17913 . . . . . . . . . . . . . . 15 Rel (𝑂 Func 𝑆)
32 1st2ndbr 8066 . . . . . . . . . . . . . . 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 17919 . . . . . . . . . . . 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 2735 . . . . . . . . . . . . 13 (Hom ‘𝐶) = (Hom ‘𝐶)
4140, 4oppchom 17761 . . . . . . . . . . . 12 (𝑋(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑋)
4239, 41eleqtrrdi 2850 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑔 ∈ (𝑋(Hom ‘𝑂)𝑧))
4338, 42ffvelcdmd 7105 . . . . . . . . . 10 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd𝐹)𝑧)‘𝑔) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
4415unssbd 4204 . . . . . . . . . . . . . 14 (𝜑𝑈𝑉)
4513, 44ssexd 5330 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ V)
4645adantr 480 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → 𝑈 ∈ V)
4746adantr 480 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑈 ∈ V)
48 eqid 2735 . . . . . . . . . . . . . . 15 (Base‘𝑆) = (Base‘𝑆)
4928, 48, 33funcf1 17917 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
505, 45setcbas 18132 . . . . . . . . . . . . . . 15 (𝜑𝑈 = (Base‘𝑆))
5150feq3d 6724 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹):𝐵𝑈 ↔ (1st𝐹):𝐵⟶(Base‘𝑆)))
5249, 51mpbird 257 . . . . . . . . . . . . 13 (𝜑 → (1st𝐹):𝐵𝑈)
5352, 17ffvelcdmd 7105 . . . . . . . . . . . 12 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
5453ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
5552ffvelcdmda 7104 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → ((1st𝐹)‘𝑧) ∈ 𝑈)
5655adantr 480 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑧) ∈ 𝑈)
575, 47, 30, 54, 56elsetchom 18135 . . . . . . . . . 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 7105 . . . . . . . 8 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴) ∈ ((1st𝐹)‘𝑧))
6160fmpttd 7135 . . . . . . 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 18321 . . . . . . . 8 ((𝜑𝑧𝐵) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) = (𝑧(Hom ‘𝐶)𝑋))
6463feq2d 6723 . . . . . . 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 18320 . . . . . . . . . . 11 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
67 1st2ndbr 8066 . . . . . . . . . . 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 17917 . . . . . . . . 9 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
7050feq3d 6724 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑋)):𝐵𝑈 ↔ (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆)))
7169, 70mpbird 257 . . . . . . . 8 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵𝑈)
7271ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑧𝐵) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ∈ 𝑈)
735, 46, 30, 72, 55elsetchom 18135 . . . . . 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 3144 . . . 4 (𝜑 → ∀𝑧𝐵 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
762fvexi 6921 . . . . 5 𝐵 ∈ V
77 mptelixpg 8974 . . . . 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 2839 . 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 1193 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑧𝐵)
841, 2, 81, 82, 40, 83yon11 18321 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) = (𝑧(Hom ‘𝐶)𝑋))
8584eleq2d 2825 . . . . . . . 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 2735 . . . . . . . . . . . 12 (comp‘𝑂) = (comp‘𝑂)
88 eqid 2735 . . . . . . . . . . . 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 1194 . . . . . . . . . . . . 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 2850 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑘 ∈ (𝑋(Hom ‘𝑂)𝑧))
97 simplr3 1216 . . . . . . . . . . . 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 17922 . . . . . . . . . . 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 2735 . . . . . . . . . . . . 13 (comp‘𝐶) = (comp‘𝐶)
1002, 99, 4, 91, 92, 94oppcco 17763 . . . . . . . . . . . 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 6911 . . . . . . . . . . 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 1187 . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . 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 17919 . . . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . 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 18135 . . . . . . . . . . . . 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 17919 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑧(2nd𝐹)𝑤):(𝑧(Hom ‘𝑂)𝑤)⟶(((1st𝐹)‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑤)))
116 simpr3 1195 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ∈ (𝑧(Hom ‘𝑂)𝑤))
117115, 116ffvelcdmd 7105 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd𝐹)𝑤)‘) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑤)))
1185, 102, 30, 105, 108elsetchom 18135 . . . . . . . . . . . . . 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 18137 . . . . . . . . . . 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 2783 . . . . . . . . . 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 6909 . . . . . . . . 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 7008 . . . . . . . . . 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 2775 . . . . . . . 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 17761 . . . . . . . . . . . 12 (𝑧(Hom ‘𝑂)𝑤) = (𝑤(Hom ‘𝐶)𝑧)
13097, 129eleqtrdi 2849 . . . . . . . . . . 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 18322 . . . . . . . . . 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 6911 . . . . . . . . 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 17730 . . . . . . . . . 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 18333 . . . . . . . . 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 2775 . . . . . . . 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 18333 . . . . . . . . 9 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘) = (((𝑋(2nd𝐹)𝑧)‘𝑘)‘𝐴))
141140fveq2d 6911 . . . . . . . 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 2785 . . . . . . 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 5248 . . . . 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 6907 . . . . . . . 8 (𝑧 = 𝑤 → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = (((𝐹𝑁𝑋)‘𝐴)‘𝑤))
146 fveq2 6907 . . . . . . . 8 (𝑧 = 𝑤 → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) = ((1st ‘((1st𝑌)‘𝑋))‘𝑤))
147 fveq2 6907 . . . . . . . 8 (𝑧 = 𝑤 → ((1st𝐹)‘𝑧) = ((1st𝐹)‘𝑤))
148145, 146, 147feq123d 6726 . . . . . . 7 (𝑧 = 𝑤 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧) ↔ (((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1st ‘((1st𝑌)‘𝑋))‘𝑤)⟶((1st𝐹)‘𝑤)))
14927fveq1d 6909 . . . . . . . . . . . 12 (𝜑 → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = ((𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))‘𝑧))
150 ovex 7464 . . . . . . . . . . . . . 14 (𝑧(Hom ‘𝐶)𝑋) ∈ V
151150mptex 7243 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ V
152 eqid 2735 . . . . . . . . . . . . . 14 (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))) = (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
153152fvmpt2 7027 . . . . . . . . . . . . 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 2795 . . . . . . . . . . 11 ((𝜑𝑧𝐵) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
156155feq1d 6721 . . . . . . . . . 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 3144 . . . . . . . 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 3623 . . . . . 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 17919 . . . . . . . 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 7105 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑤)))
164723ad2antr1 1187 . . . . . . . 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 7105 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st ‘((1st𝑌)‘𝑋))‘𝑤) ∈ 𝑈)
1675, 102, 30, 164, 166elsetchom 18135 . . . . . . 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 7153 . . . . . 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 1187 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
172 fcompt 7153 . . . . . 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 2785 . . . 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 18137 . . . 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 18137 . . . 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 2785 . . 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 3203 . 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 2735 . . 3 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
180179, 28, 29, 30, 88, 66, 16isnat2 18003 . 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 1537  wcel 2106  wral 3059  Vcvv 3478  cun 3961  wss 3963  cop 4637   class class class wbr 5148  cmpt 5231  ran crn 5690  ccom 5693  Rel wrel 5694  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  tpos ctpos 8249  Xcixp 8936  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Idccid 17710  Homf chomf 17711  oppCatcoppc 17756   Func cfunc 17905  func ccofu 17907   Nat cnat 17996   FuncCat cfuc 17997  SetCatcsetc 18129   ×c cxpc 18224   1stF c1stf 18225   2ndF c2ndf 18226   ⟨,⟩F cprf 18227   evalF cevlf 18266  HomFchof 18305  Yoncyon 18306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-hom 17322  df-cco 17323  df-cat 17713  df-cid 17714  df-homf 17715  df-comf 17716  df-oppc 17757  df-func 17909  df-nat 17998  df-fuc 17999  df-setc 18130  df-xpc 18228  df-curf 18271  df-hof 18307  df-yon 18308
This theorem is referenced by:  yonedainv  18338
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