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Theorem yonedalem4c 17356
Description: Lemma for yoneda 17362. (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 17354 . . . 4 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))))
21 oveq1 7023 . . . . . 6 (𝑦 = 𝑧 → (𝑦(Hom ‘𝐶)𝑋) = (𝑧(Hom ‘𝐶)𝑋))
22 oveq2 7024 . . . . . . . 8 (𝑦 = 𝑧 → (𝑋(2nd𝐹)𝑦) = (𝑋(2nd𝐹)𝑧))
2322fveq1d 6540 . . . . . . 7 (𝑦 = 𝑧 → ((𝑋(2nd𝐹)𝑦)‘𝑔) = ((𝑋(2nd𝐹)𝑧)‘𝑔))
2423fveq1d 6540 . . . . . 6 (𝑦 = 𝑧 → (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴) = (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))
2521, 24mpteq12dv 5045 . . . . 5 (𝑦 = 𝑧 → (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴)) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
2625cbvmptv 5061 . . . 4 (𝑦𝐵 ↦ (𝑔 ∈ (𝑦(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑦)‘𝑔)‘𝐴))) = (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
2720, 26syl6eq 2847 . . 3 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) = (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))))
284, 2oppcbas 16817 . . . . . . . . . . . . 13 𝐵 = (Base‘𝑂)
29 eqid 2795 . . . . . . . . . . . . 13 (Hom ‘𝑂) = (Hom ‘𝑂)
30 eqid 2795 . . . . . . . . . . . . 13 (Hom ‘𝑆) = (Hom ‘𝑆)
31 relfunc 16961 . . . . . . . . . . . . . . 15 Rel (𝑂 Func 𝑆)
32 1st2ndbr 7597 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
3517adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝑋𝐵)
36 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑧𝐵) → 𝑧𝐵)
3728, 29, 30, 34, 35, 36funcf2 16967 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → (𝑋(2nd𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
3837adantr 481 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (𝑋(2nd𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
39 simpr 485 . . . . . . . . . . . 12 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋))
40 eqid 2795 . . . . . . . . . . . . 13 (Hom ‘𝐶) = (Hom ‘𝐶)
4140, 4oppchom 16814 . . . . . . . . . . . 12 (𝑋(Hom ‘𝑂)𝑧) = (𝑧(Hom ‘𝐶)𝑋)
4239, 41syl6eleqr 2894 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑔 ∈ (𝑋(Hom ‘𝑂)𝑧))
4338, 42ffvelrnd 6717 . . . . . . . . . 10 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd𝐹)𝑧)‘𝑔) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
4415unssbd 4085 . . . . . . . . . . . . . 14 (𝜑𝑈𝑉)
4513, 44ssexd 5119 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ V)
4645adantr 481 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → 𝑈 ∈ V)
4746adantr 481 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑈 ∈ V)
48 eqid 2795 . . . . . . . . . . . . . . 15 (Base‘𝑆) = (Base‘𝑆)
4928, 48, 33funcf1 16965 . . . . . . . . . . . . . 14 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝑆))
505, 45setcbas 17167 . . . . . . . . . . . . . . 15 (𝜑𝑈 = (Base‘𝑆))
5150feq3d 6369 . . . . . . . . . . . . . 14 (𝜑 → ((1st𝐹):𝐵𝑈 ↔ (1st𝐹):𝐵⟶(Base‘𝑆)))
5249, 51mpbird 258 . . . . . . . . . . . . 13 (𝜑 → (1st𝐹):𝐵𝑈)
5352, 17ffvelrnd 6717 . . . . . . . . . . . 12 (𝜑 → ((1st𝐹)‘𝑋) ∈ 𝑈)
5453ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
5552ffvelrnda 6716 . . . . . . . . . . . 12 ((𝜑𝑧𝐵) → ((1st𝐹)‘𝑧) ∈ 𝑈)
5655adantr 481 . . . . . . . . . . 11 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑧) ∈ 𝑈)
575, 47, 30, 54, 56elsetchom 17170 . . . . . . . . . 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 233 . . . . . . . . 9 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((𝑋(2nd𝐹)𝑧)‘𝑔):((1st𝐹)‘𝑋)⟶((1st𝐹)‘𝑧))
5919ad2antrr 722 . . . . . . . . 9 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐴 ∈ ((1st𝐹)‘𝑋))
6058, 59ffvelrnd 6717 . . . . . . . 8 (((𝜑𝑧𝐵) ∧ 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴) ∈ ((1st𝐹)‘𝑧))
6160fmpttd 6742 . . . . . . 7 ((𝜑𝑧𝐵) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)):(𝑧(Hom ‘𝐶)𝑋)⟶((1st𝐹)‘𝑧))
6212adantr 481 . . . . . . . . 9 ((𝜑𝑧𝐵) → 𝐶 ∈ Cat)
631, 2, 62, 35, 40, 36yon11 17343 . . . . . . . 8 ((𝜑𝑧𝐵) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) = (𝑧(Hom ‘𝐶)𝑋))
6463feq2d 6368 . . . . . . 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 258 . . . . . 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 17342 . . . . . . . . . . 11 (𝜑 → ((1st𝑌)‘𝑋) ∈ (𝑂 Func 𝑆))
67 1st2ndbr 7597 . . . . . . . . . . 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 16965 . . . . . . . . 9 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆))
7050feq3d 6369 . . . . . . . . 9 (𝜑 → ((1st ‘((1st𝑌)‘𝑋)):𝐵𝑈 ↔ (1st ‘((1st𝑌)‘𝑋)):𝐵⟶(Base‘𝑆)))
7169, 70mpbird 258 . . . . . . . 8 (𝜑 → (1st ‘((1st𝑌)‘𝑋)):𝐵𝑈)
7271ffvelrnda 6716 . . . . . . 7 ((𝜑𝑧𝐵) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ∈ 𝑈)
735, 46, 30, 72, 55elsetchom 17170 . . . . . 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 258 . . . . 5 ((𝜑𝑧𝐵) → (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
7574ralrimiva 3149 . . . 4 (𝜑 → ∀𝑧𝐵 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
762fvexi 6552 . . . . 5 𝐵 ∈ V
77 mptelixpg 8347 . . . . 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 235 . . 3 (𝜑 → (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))) ∈ X𝑧𝐵 (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
8027, 79eqeltrd 2883 . 2 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ X𝑧𝐵 (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
8112adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝐶 ∈ Cat)
8217adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑋𝐵)
83 simpr1 1187 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑧𝐵)
841, 2, 81, 82, 40, 83yon11 17343 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) = (𝑧(Hom ‘𝐶)𝑋))
8584eleq2d 2868 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑘 ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ↔ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)))
8685biimpa 477 . . . . . . 7 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ ((1st ‘((1st𝑌)‘𝑋))‘𝑧)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋))
87 eqid 2795 . . . . . . . . . . . 12 (comp‘𝑂) = (comp‘𝑂)
88 eqid 2795 . . . . . . . . . . . 12 (comp‘𝑆) = (comp‘𝑆)
8933adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
9089adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (1st𝐹)(𝑂 Func 𝑆)(2nd𝐹))
9182adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑋𝐵)
9283adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑧𝐵)
93 simpr2 1188 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑤𝐵)
9493adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑤𝐵)
95 simpr 485 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋))
9695, 41syl6eleqr 2894 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑘 ∈ (𝑋(Hom ‘𝑂)𝑧))
97 simplr3 1210 . . . . . . . . . . . 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 16970 . . . . . . . . . . 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 2795 . . . . . . . . . . . . 13 (comp‘𝐶) = (comp‘𝐶)
1002, 99, 4, 91, 92, 94oppcco 16816 . . . . . . . . . . . 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 6542 . . . . . . . . . . 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 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → 𝑈 ∈ V)
103102adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑈 ∈ V)
10453ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑋) ∈ 𝑈)
105553ad2antr1 1181 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st𝐹)‘𝑧) ∈ 𝑈)
106105adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑧) ∈ 𝑈)
10752adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st𝐹):𝐵𝑈)
108107, 93ffvelrnd 6717 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st𝐹)‘𝑤) ∈ 𝑈)
109108adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((1st𝐹)‘𝑤) ∈ 𝑈)
11028, 29, 30, 89, 82, 83funcf2 16967 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑋(2nd𝐹)𝑧):(𝑋(Hom ‘𝑂)𝑧)⟶(((1st𝐹)‘𝑋)(Hom ‘𝑆)((1st𝐹)‘𝑧)))
111110adantr 481 . . . . . . . . . . . . . 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, 96ffvelrnd 6717 . . . . . . . . . . . . 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 17170 . . . . . . . . . . . . 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 233 . . . . . . . . . . . 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 16967 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (𝑧(2nd𝐹)𝑤):(𝑧(Hom ‘𝑂)𝑤)⟶(((1st𝐹)‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑤)))
116 simpr3 1189 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ∈ (𝑧(Hom ‘𝑂)𝑤))
117115, 116ffvelrnd 6717 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd𝐹)𝑤)‘) ∈ (((1st𝐹)‘𝑧)(Hom ‘𝑆)((1st𝐹)‘𝑤)))
1185, 102, 30, 105, 108elsetchom 17170 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd𝐹)𝑤)‘):((1st𝐹)‘𝑧)⟶((1st𝐹)‘𝑤))
120119adantr 481 . . . . . . . . . . . 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 17172 . . . . . . . . . . 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 2839 . . . . . . . . . 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 6540 . . . . . . . . 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 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐴 ∈ ((1st𝐹)‘𝑋))
125 fvco3 6627 . . . . . . . . . 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 2831 . . . . . . . 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 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝐶 ∈ Cat)
12940, 4oppchom 16814 . . . . . . . . . . . 12 (𝑧(Hom ‘𝑂)𝑤) = (𝑤(Hom ‘𝐶)𝑧)
13097, 129syl6eleq 2893 . . . . . . . . . . 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 17344 . . . . . . . . . 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 6542 . . . . . . . . 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 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → 𝑉𝑊)
13414ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ran (Homf𝐶) ⊆ 𝑈)
13515ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → (ran (Homf𝑄) ∪ 𝑈) ⊆ 𝑉)
13616ad2antrr 722 . . . . . . . . . 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 16785 . . . . . . . . . 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 17355 . . . . . . . . 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 2831 . . . . . . . 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 17355 . . . . . . . . 9 (((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) ∧ 𝑘 ∈ (𝑧(Hom ‘𝐶)𝑋)) → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧)‘𝑘) = (((𝑋(2nd𝐹)𝑧)‘𝑘)‘𝐴))
141140fveq2d 6542 . . . . . . . 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 2841 . . . . . . 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 5055 . . . . 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 6538 . . . . . . . 8 (𝑧 = 𝑤 → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = (((𝐹𝑁𝑋)‘𝐴)‘𝑤))
146 fveq2 6538 . . . . . . . 8 (𝑧 = 𝑤 → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) = ((1st ‘((1st𝑌)‘𝑋))‘𝑤))
147 fveq2 6538 . . . . . . . 8 (𝑧 = 𝑤 → ((1st𝐹)‘𝑧) = ((1st𝐹)‘𝑤))
148145, 146, 147feq123d 6371 . . . . . . 7 (𝑧 = 𝑤 → ((((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧) ↔ (((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1st ‘((1st𝑌)‘𝑋))‘𝑤)⟶((1st𝐹)‘𝑤)))
14927fveq1d 6540 . . . . . . . . . . . 12 (𝜑 → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = ((𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))‘𝑧))
150 ovex 7048 . . . . . . . . . . . . . 14 (𝑧(Hom ‘𝐶)𝑋) ∈ V
151150mptex 6852 . . . . . . . . . . . . 13 (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)) ∈ V
152 eqid 2795 . . . . . . . . . . . . . 14 (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴))) = (𝑧𝐵 ↦ (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
153152fvmpt2 6645 . . . . . . . . . . . . 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 687 . . . . . . . . . . . 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 2851 . . . . . . . . . . 11 ((𝜑𝑧𝐵) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧) = (𝑔 ∈ (𝑧(Hom ‘𝐶)𝑋) ↦ (((𝑋(2nd𝐹)𝑧)‘𝑔)‘𝐴)))
156155feq1d 6367 . . . . . . . . . 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 258 . . . . . . . . 9 ((𝜑𝑧𝐵) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
158157ralrimiva 3149 . . . . . . . 8 (𝜑 → ∀𝑧𝐵 (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
159158adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ∀𝑧𝐵 (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
160148, 159, 93rspcdva 3565 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝐹𝑁𝑋)‘𝐴)‘𝑤):((1st ‘((1st𝑌)‘𝑋))‘𝑤)⟶((1st𝐹)‘𝑤))
16168adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st ‘((1st𝑌)‘𝑋))(𝑂 Func 𝑆)(2nd ‘((1st𝑌)‘𝑋)))
16228, 29, 30, 161, 83, 93funcf2 16967 . . . . . . . 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, 116ffvelrnd 6717 . . . . . . 7 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘) ∈ (((1st ‘((1st𝑌)‘𝑋))‘𝑧)(Hom ‘𝑆)((1st ‘((1st𝑌)‘𝑋))‘𝑤)))
164723ad2antr1 1181 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st ‘((1st𝑌)‘𝑋))‘𝑧) ∈ 𝑈)
16571adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (1st ‘((1st𝑌)‘𝑋)):𝐵𝑈)
166165, 93ffvelrnd 6717 . . . . . . . 8 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((1st ‘((1st𝑌)‘𝑋))‘𝑤) ∈ 𝑈)
1675, 102, 30, 164, 166elsetchom 17170 . . . . . . 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 233 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → ((𝑧(2nd ‘((1st𝑌)‘𝑋))𝑤)‘):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st ‘((1st𝑌)‘𝑋))‘𝑤))
169 fcompt 6758 . . . . . 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 1181 . . . . . 6 ((𝜑 ∧ (𝑧𝐵𝑤𝐵 ∈ (𝑧(Hom ‘𝑂)𝑤))) → (((𝐹𝑁𝑋)‘𝐴)‘𝑧):((1st ‘((1st𝑌)‘𝑋))‘𝑧)⟶((1st𝐹)‘𝑧))
172 fcompt 6758 . . . . . 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 2841 . . . 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 17172 . . . 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 17172 . . . 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 2841 . . 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 3159 . 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 2795 . . 3 (𝑂 Nat 𝑆) = (𝑂 Nat 𝑆)
180179, 28, 29, 30, 88, 66, 16isnat2 17047 . 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 709 1 (𝜑 → ((𝐹𝑁𝑋)‘𝐴) ∈ (((1st𝑌)‘𝑋)(𝑂 Nat 𝑆)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  Vcvv 3437  cun 3857  wss 3859  cop 4478   class class class wbr 4962  cmpt 5041  ran crn 5444  ccom 5447  Rel wrel 5448  wf 6221  cfv 6225  (class class class)co 7016  cmpo 7018  1st c1st 7543  2nd c2nd 7544  tpos ctpos 7742  Xcixp 8310  Basecbs 16312  Hom chom 16405  compcco 16406  Catccat 16764  Idccid 16765  Homf chomf 16766  oppCatcoppc 16810   Func cfunc 16953  func ccofu 16955   Nat cnat 17040   FuncCat cfuc 17041  SetCatcsetc 17164   ×c cxpc 17247   1stF c1stf 17248   2ndF c2ndf 17249   ⟨,⟩F cprf 17250   evalF cevlf 17288  HomFchof 17327  Yoncyon 17328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-tpos 7743  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-fz 12743  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-hom 16418  df-cco 16419  df-cat 16768  df-cid 16769  df-homf 16770  df-comf 16771  df-oppc 16811  df-func 16957  df-nat 17042  df-fuc 17043  df-setc 17165  df-xpc 17251  df-curf 17293  df-hof 17329  df-yon 17330
This theorem is referenced by:  yonedainv  17360
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