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Theorem fucpropd 18034
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same functor categories. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
fucpropd.1 (𝜑 → (Homf𝐴) = (Homf𝐵))
fucpropd.2 (𝜑 → (compf𝐴) = (compf𝐵))
fucpropd.3 (𝜑 → (Homf𝐶) = (Homf𝐷))
fucpropd.4 (𝜑 → (compf𝐶) = (compf𝐷))
fucpropd.a (𝜑𝐴 ∈ Cat)
fucpropd.b (𝜑𝐵 ∈ Cat)
fucpropd.c (𝜑𝐶 ∈ Cat)
fucpropd.d (𝜑𝐷 ∈ Cat)
Assertion
Ref Expression
fucpropd (𝜑 → (𝐴 FuncCat 𝐶) = (𝐵 FuncCat 𝐷))

Proof of Theorem fucpropd
Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucpropd.1 . . . . 5 (𝜑 → (Homf𝐴) = (Homf𝐵))
2 fucpropd.2 . . . . 5 (𝜑 → (compf𝐴) = (compf𝐵))
3 fucpropd.3 . . . . 5 (𝜑 → (Homf𝐶) = (Homf𝐷))
4 fucpropd.4 . . . . 5 (𝜑 → (compf𝐶) = (compf𝐷))
5 fucpropd.a . . . . 5 (𝜑𝐴 ∈ Cat)
6 fucpropd.b . . . . 5 (𝜑𝐵 ∈ Cat)
7 fucpropd.c . . . . 5 (𝜑𝐶 ∈ Cat)
8 fucpropd.d . . . . 5 (𝜑𝐷 ∈ Cat)
91, 2, 3, 4, 5, 6, 7, 8funcpropd 17954 . . . 4 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
109opeq2d 4885 . . 3 (𝜑 → ⟨(Base‘ndx), (𝐴 Func 𝐶)⟩ = ⟨(Base‘ndx), (𝐵 Func 𝐷)⟩)
111, 2, 3, 4, 5, 6, 7, 8natpropd 18033 . . . 4 (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
1211opeq2d 4885 . . 3 (𝜑 → ⟨(Hom ‘ndx), (𝐴 Nat 𝐶)⟩ = ⟨(Hom ‘ndx), (𝐵 Nat 𝐷)⟩)
139sqxpeqd 5721 . . . . 5 (𝜑 → ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) = ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)))
149adantr 480 . . . . 5 ((𝜑𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶))) → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
15 nfv 1912 . . . . . 6 𝑓(𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶)))
16 nfcsb1v 3933 . . . . . . 7 𝑓(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥))))
1716a1i 11 . . . . . 6 ((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) → 𝑓(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))
18 fvexd 6922 . . . . . 6 ((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) → (1st𝑣) ∈ V)
19 nfv 1912 . . . . . . . 8 𝑔((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣))
20 nfcsb1v 3933 . . . . . . . . 9 𝑔(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥))))
2120a1i 11 . . . . . . . 8 (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) → 𝑔(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))
22 fvexd 6922 . . . . . . . 8 (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) → (2nd𝑣) ∈ V)
2311ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
2423oveqd 7448 . . . . . . . . . 10 ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) → (𝑔(𝐴 Nat 𝐶)) = (𝑔(𝐵 Nat 𝐷)))
2523oveqdr 7459 . . . . . . . . . 10 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ 𝑏 ∈ (𝑔(𝐴 Nat 𝐶))) → (𝑓(𝐴 Nat 𝐶)𝑔) = (𝑓(𝐵 Nat 𝐷)𝑔))
261homfeqbas 17741 . . . . . . . . . . . 12 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
2726ad4antr 732 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (Base‘𝐴) = (Base‘𝐵))
28 eqid 2735 . . . . . . . . . . . 12 (Base‘𝐶) = (Base‘𝐶)
29 eqid 2735 . . . . . . . . . . . 12 (Hom ‘𝐶) = (Hom ‘𝐶)
30 eqid 2735 . . . . . . . . . . . 12 (comp‘𝐶) = (comp‘𝐶)
31 eqid 2735 . . . . . . . . . . . 12 (comp‘𝐷) = (comp‘𝐷)
323ad5antr 734 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → (Homf𝐶) = (Homf𝐷))
334ad5antr 734 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → (compf𝐶) = (compf𝐷))
34 eqid 2735 . . . . . . . . . . . . . 14 (Base‘𝐴) = (Base‘𝐴)
35 relfunc 17913 . . . . . . . . . . . . . . 15 Rel (𝐴 Func 𝐶)
36 simpllr 776 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → 𝑓 = (1st𝑣))
37 simp-4r 784 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶)))
3837simpld 494 . . . . . . . . . . . . . . . . 17 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → 𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)))
39 xp1st 8045 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) → (1st𝑣) ∈ (𝐴 Func 𝐶))
4038, 39syl 17 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st𝑣) ∈ (𝐴 Func 𝐶))
4136, 40eqeltrd 2839 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → 𝑓 ∈ (𝐴 Func 𝐶))
42 1st2ndbr 8066 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
4335, 41, 42sylancr 587 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
4434, 28, 43funcf1 17917 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st𝑓):(Base‘𝐴)⟶(Base‘𝐶))
4544ffvelcdmda 7104 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → ((1st𝑓)‘𝑥) ∈ (Base‘𝐶))
46 simplr 769 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → 𝑔 = (2nd𝑣))
47 xp2nd 8046 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) → (2nd𝑣) ∈ (𝐴 Func 𝐶))
4838, 47syl 17 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (2nd𝑣) ∈ (𝐴 Func 𝐶))
4946, 48eqeltrd 2839 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → 𝑔 ∈ (𝐴 Func 𝐶))
50 1st2ndbr 8066 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
5135, 49, 50sylancr 587 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
5234, 28, 51funcf1 17917 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st𝑔):(Base‘𝐴)⟶(Base‘𝐶))
5352ffvelcdmda 7104 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → ((1st𝑔)‘𝑥) ∈ (Base‘𝐶))
5437simprd 495 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → ∈ (𝐴 Func 𝐶))
55 1st2ndbr 8066 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ ∈ (𝐴 Func 𝐶)) → (1st)(𝐴 Func 𝐶)(2nd))
5635, 54, 55sylancr 587 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st)(𝐴 Func 𝐶)(2nd))
5734, 28, 56funcf1 17917 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (1st):(Base‘𝐴)⟶(Base‘𝐶))
5857ffvelcdmda 7104 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → ((1st)‘𝑥) ∈ (Base‘𝐶))
59 eqid 2735 . . . . . . . . . . . . 13 (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
60 simplrr 778 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))
6159, 60nat1st2nd 18006 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑎 ∈ (⟨(1st𝑓), (2nd𝑓)⟩(𝐴 Nat 𝐶)⟨(1st𝑔), (2nd𝑔)⟩))
62 simpr 484 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑥 ∈ (Base‘𝐴))
6359, 61, 34, 29, 62natcl 18008 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑎𝑥) ∈ (((1st𝑓)‘𝑥)(Hom ‘𝐶)((1st𝑔)‘𝑥)))
64 simplrl 777 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑏 ∈ (𝑔(𝐴 Nat 𝐶)))
6559, 64nat1st2nd 18006 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → 𝑏 ∈ (⟨(1st𝑔), (2nd𝑔)⟩(𝐴 Nat 𝐶)⟨(1st), (2nd)⟩))
6659, 65, 34, 29, 62natcl 18008 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → (𝑏𝑥) ∈ (((1st𝑔)‘𝑥)(Hom ‘𝐶)((1st)‘𝑥)))
6728, 29, 30, 31, 32, 33, 45, 53, 58, 63, 66comfeqval 17753 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) ∧ 𝑥 ∈ (Base‘𝐴)) → ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐶)((1st)‘𝑥))(𝑎𝑥)) = ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))
6827, 67mpteq12dva 5237 . . . . . . . . . 10 (((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) ∧ (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)) ∧ 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))) → (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐶)((1st)‘𝑥))(𝑎𝑥))) = (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥))))
6924, 25, 68mpoeq123dva 7507 . . . . . . . . 9 ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) → (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐶)((1st)‘𝑥))(𝑎𝑥)))) = (𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))
70 csbeq1a 3922 . . . . . . . . . 10 (𝑔 = (2nd𝑣) → (𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))) = (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))
7170adantl 481 . . . . . . . . 9 ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) → (𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))) = (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))
7269, 71eqtrd 2775 . . . . . . . 8 ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) → (𝑏 ∈ (𝑔(𝐴 Nat 𝐶)), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐶)((1st)‘𝑥))(𝑎𝑥)))) = (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))
7319, 21, 22, 72csbiedf 3939 . . . . . . 7 (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) → (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐶)((1st)‘𝑥))(𝑎𝑥)))) = (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))
74 csbeq1a 3922 . . . . . . . 8 (𝑓 = (1st𝑣) → (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))) = (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))
7574adantl 481 . . . . . . 7 (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) → (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))) = (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))
7673, 75eqtrd 2775 . . . . . 6 (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) → (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐶)((1st)‘𝑥))(𝑎𝑥)))) = (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))
7715, 17, 18, 76csbiedf 3939 . . . . 5 ((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) → (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐶)((1st)‘𝑥))(𝑎𝑥)))) = (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))
7813, 14, 77mpoeq123dva 7507 . . . 4 (𝜑 → (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)), ∈ (𝐴 Func 𝐶) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐶)((1st)‘𝑥))(𝑎𝑥))))) = (𝑣 ∈ ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)), ∈ (𝐵 Func 𝐷) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥))))))
7978opeq2d 4885 . . 3 (𝜑 → ⟨(comp‘ndx), (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)), ∈ (𝐴 Func 𝐶) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐶)((1st)‘𝑥))(𝑎𝑥)))))⟩ = ⟨(comp‘ndx), (𝑣 ∈ ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)), ∈ (𝐵 Func 𝐷) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))⟩)
8010, 12, 79tpeq123d 4753 . 2 (𝜑 → {⟨(Base‘ndx), (𝐴 Func 𝐶)⟩, ⟨(Hom ‘ndx), (𝐴 Nat 𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)), ∈ (𝐴 Func 𝐶) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐶)((1st)‘𝑥))(𝑎𝑥)))))⟩} = {⟨(Base‘ndx), (𝐵 Func 𝐷)⟩, ⟨(Hom ‘ndx), (𝐵 Nat 𝐷)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)), ∈ (𝐵 Func 𝐷) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))⟩})
81 eqid 2735 . . 3 (𝐴 FuncCat 𝐶) = (𝐴 FuncCat 𝐶)
82 eqid 2735 . . 3 (𝐴 Func 𝐶) = (𝐴 Func 𝐶)
83 eqidd 2736 . . 3 (𝜑 → (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)), ∈ (𝐴 Func 𝐶) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐶)((1st)‘𝑥))(𝑎𝑥))))) = (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)), ∈ (𝐴 Func 𝐶) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐶)((1st)‘𝑥))(𝑎𝑥))))))
8481, 82, 59, 34, 30, 5, 7, 83fucval 18014 . 2 (𝜑 → (𝐴 FuncCat 𝐶) = {⟨(Base‘ndx), (𝐴 Func 𝐶)⟩, ⟨(Hom ‘ndx), (𝐴 Nat 𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)), ∈ (𝐴 Func 𝐶) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐴 Nat 𝐶)), 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔) ↦ (𝑥 ∈ (Base‘𝐴) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐶)((1st)‘𝑥))(𝑎𝑥)))))⟩})
85 eqid 2735 . . 3 (𝐵 FuncCat 𝐷) = (𝐵 FuncCat 𝐷)
86 eqid 2735 . . 3 (𝐵 Func 𝐷) = (𝐵 Func 𝐷)
87 eqid 2735 . . 3 (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
88 eqid 2735 . . 3 (Base‘𝐵) = (Base‘𝐵)
89 eqidd 2736 . . 3 (𝜑 → (𝑣 ∈ ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)), ∈ (𝐵 Func 𝐷) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥))))) = (𝑣 ∈ ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)), ∈ (𝐵 Func 𝐷) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥))))))
9085, 86, 87, 88, 31, 6, 8, 89fucval 18014 . 2 (𝜑 → (𝐵 FuncCat 𝐷) = {⟨(Base‘ndx), (𝐵 Func 𝐷)⟩, ⟨(Hom ‘ndx), (𝐵 Nat 𝐷)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)), ∈ (𝐵 Func 𝐷) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝐵 Nat 𝐷)), 𝑎 ∈ (𝑓(𝐵 Nat 𝐷)𝑔) ↦ (𝑥 ∈ (Base‘𝐵) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝐷)((1st)‘𝑥))(𝑎𝑥)))))⟩})
9180, 84, 903eqtr4d 2785 1 (𝜑 → (𝐴 FuncCat 𝐶) = (𝐵 FuncCat 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wnfc 2888  Vcvv 3478  csb 3908  {ctp 4635  cop 4637   class class class wbr 5148  cmpt 5231   × cxp 5687  Rel wrel 5694  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  ndxcnx 17227  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Homf chomf 17711  compfccomf 17712   Func cfunc 17905   Nat cnat 17996   FuncCat cfuc 17997
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3060  df-rex 3069  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-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-id 5583  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-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-1st 8013  df-2nd 8014  df-map 8867  df-ixp 8937  df-cat 17713  df-cid 17714  df-homf 17715  df-comf 17716  df-func 17909  df-nat 17998  df-fuc 17999
This theorem is referenced by:  oyoncl  18327
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