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Theorem fucpropd 18025
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 17947 . . . 4 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
109opeq2d 4880 . . 3 (𝜑 → ⟨(Base‘ndx), (𝐴 Func 𝐶)⟩ = ⟨(Base‘ndx), (𝐵 Func 𝐷)⟩)
111, 2, 3, 4, 5, 6, 7, 8natpropd 18024 . . . 4 (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
1211opeq2d 4880 . . 3 (𝜑 → ⟨(Hom ‘ndx), (𝐴 Nat 𝐶)⟩ = ⟨(Hom ‘ndx), (𝐵 Nat 𝐷)⟩)
139sqxpeqd 5717 . . . . 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 1914 . . . . . 6 𝑓(𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶)))
16 nfcsb1v 3923 . . . . . . 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 6921 . . . . . 6 ((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) → (1st𝑣) ∈ V)
19 nfv 1914 . . . . . . . 8 𝑔((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣))
20 nfcsb1v 3923 . . . . . . . . 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 6921 . . . . . . . 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 17739 . . . . . . . . . . . 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 2737 . . . . . . . . . . . 12 (Base‘𝐶) = (Base‘𝐶)
29 eqid 2737 . . . . . . . . . . . 12 (Hom ‘𝐶) = (Hom ‘𝐶)
30 eqid 2737 . . . . . . . . . . . 12 (comp‘𝐶) = (comp‘𝐶)
31 eqid 2737 . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . 14 (Base‘𝐴) = (Base‘𝐴)
35 relfunc 17907 . . . . . . . . . . . . . . 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 8046 . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . 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 8067 . . . . . . . . . . . . . . 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 17911 . . . . . . . . . . . . 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 8047 . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . 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 8067 . . . . . . . . . . . . . . 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 17911 . . . . . . . . . . . . 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 8067 . . . . . . . . . . . . . . 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 17911 . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . 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 17999 . . . . . . . . . . . . 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 18001 . . . . . . . . . . . 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 17999 . . . . . . . . . . . . 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 18001 . . . . . . . . . . . 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 17751 . . . . . . . . . . 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 5231 . . . . . . . . . 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 3913 . . . . . . . . . 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 2777 . . . . . . . 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 3929 . . . . . . 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 3913 . . . . . . . 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 2777 . . . . . 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 3929 . . . . 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 4880 . . 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 4748 . 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 2737 . . 3 (𝐴 FuncCat 𝐶) = (𝐴 FuncCat 𝐶)
82 eqid 2737 . . 3 (𝐴 Func 𝐶) = (𝐴 Func 𝐶)
83 eqidd 2738 . . 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 18006 . 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 2737 . . 3 (𝐵 FuncCat 𝐷) = (𝐵 FuncCat 𝐷)
86 eqid 2737 . . 3 (𝐵 Func 𝐷) = (𝐵 Func 𝐷)
87 eqid 2737 . . 3 (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
88 eqid 2737 . . 3 (Base‘𝐵) = (Base‘𝐵)
89 eqidd 2738 . . 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 18006 . 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 2787 1 (𝜑 → (𝐴 FuncCat 𝐶) = (𝐵 FuncCat 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wnfc 2890  Vcvv 3480  csb 3899  {ctp 4630  cop 4632   class class class wbr 5143  cmpt 5225   × cxp 5683  Rel wrel 5690  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  ndxcnx 17230  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Homf chomf 17709  compfccomf 17710   Func cfunc 17899   Nat cnat 17989   FuncCat cfuc 17990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ixp 8938  df-cat 17711  df-cid 17712  df-homf 17713  df-comf 17714  df-func 17903  df-nat 17991  df-fuc 17992
This theorem is referenced by:  oyoncl  18315
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