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Theorem fucpropd 16684
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 16607 . . . 4 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
109opeq2d 4440 . . 3 (𝜑 → ⟨(Base‘ndx), (𝐴 Func 𝐶)⟩ = ⟨(Base‘ndx), (𝐵 Func 𝐷)⟩)
111, 2, 3, 4, 5, 6, 7, 8natpropd 16683 . . . 4 (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
1211opeq2d 4440 . . 3 (𝜑 → ⟨(Hom ‘ndx), (𝐴 Nat 𝐶)⟩ = ⟨(Hom ‘ndx), (𝐵 Nat 𝐷)⟩)
139sqxpeqd 5175 . . . . 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 1883 . . . . . 6 𝑓(𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶)))
16 nfcsb1v 3582 . . . . . . 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 6241 . . . . . 6 ((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) → (1st𝑣) ∈ V)
19 nfv 1883 . . . . . . . 8 𝑔((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣))
20 nfcsb1v 3582 . . . . . . . . 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 6241 . . . . . . . 8 (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) → (2nd𝑣) ∈ V)
2311ad3antrrr 766 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
2423oveqd 6707 . . . . . . . . . 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 6714 . . . . . . . . . 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 16403 . . . . . . . . . . . 12 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
2726ad4antr 769 . . . . . . . . . . 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 2651 . . . . . . . . . . . 12 (Base‘𝐶) = (Base‘𝐶)
29 eqid 2651 . . . . . . . . . . . 12 (Hom ‘𝐶) = (Hom ‘𝐶)
30 eqid 2651 . . . . . . . . . . . 12 (comp‘𝐶) = (comp‘𝐶)
31 eqid 2651 . . . . . . . . . . . 12 (comp‘𝐷) = (comp‘𝐷)
323ad5antr 773 . . . . . . . . . . . 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 773 . . . . . . . . . . . 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 2651 . . . . . . . . . . . . . 14 (Base‘𝐴) = (Base‘𝐴)
35 relfunc 16569 . . . . . . . . . . . . . . 15 Rel (𝐴 Func 𝐶)
36 simpllr 815 . . . . . . . . . . . . . . . 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 824 . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . 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 7242 . . . . . . . . . . . . . . . . 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 2730 . . . . . . . . . . . . . . 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 7261 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
4335, 41, 42sylancr 696 . . . . . . . . . . . . . 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 16573 . . . . . . . . . . . . 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))
4544ffvelrnda 6399 . . . . . . . . . . . 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 807 . . . . . . . . . . . . . . . 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 7243 . . . . . . . . . . . . . . . . 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 2730 . . . . . . . . . . . . . . 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 7261 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
5135, 49, 50sylancr 696 . . . . . . . . . . . . . 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 16573 . . . . . . . . . . . . 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))
5352ffvelrnda 6399 . . . . . . . . . . . 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 478 . . . . . . . . . . . . . . 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 7261 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ ∈ (𝐴 Func 𝐶)) → (1st)(𝐴 Func 𝐶)(2nd))
5635, 54, 55sylancr 696 . . . . . . . . . . . . . 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 16573 . . . . . . . . . . . . 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))
5857ffvelrnda 6399 . . . . . . . . . . . 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 2651 . . . . . . . . . . . . 13 (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
60 simplrr 818 . . . . . . . . . . . . . 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 16658 . . . . . . . . . . . . 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 476 . . . . . . . . . . . . 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 16660 . . . . . . . . . . . 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 817 . . . . . . . . . . . . . 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 16658 . . . . . . . . . . . . 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 16660 . . . . . . . . . . . 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 16415 . . . . . . . . . . 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 4765 . . . . . . . . . 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, 68mpt2eq123dva 6758 . . . . . . . . 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 3575 . . . . . . . . . 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 2685 . . . . . . . 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 3587 . . . . . . 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 3575 . . . . . . . 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 2685 . . . . . 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 3587 . . . . 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, 77mpt2eq123dva 6758 . . . 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 4440 . . 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 4315 . 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 2651 . . 3 (𝐴 FuncCat 𝐶) = (𝐴 FuncCat 𝐶)
82 eqid 2651 . . 3 (𝐴 Func 𝐶) = (𝐴 Func 𝐶)
83 eqidd 2652 . . 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 16665 . 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 2651 . . 3 (𝐵 FuncCat 𝐷) = (𝐵 FuncCat 𝐷)
86 eqid 2651 . . 3 (𝐵 Func 𝐷) = (𝐵 Func 𝐷)
87 eqid 2651 . . 3 (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
88 eqid 2651 . . 3 (Base‘𝐵) = (Base‘𝐵)
89 eqidd 2652 . . 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 16665 . 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 2695 1 (𝜑 → (𝐴 FuncCat 𝐶) = (𝐵 FuncCat 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wnfc 2780  Vcvv 3231  csb 3566  {ctp 4214  cop 4216   class class class wbr 4685  cmpt 4762   × cxp 5141  Rel wrel 5148  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  ndxcnx 15901  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Homf chomf 16374  compfccomf 16375   Func cfunc 16561   Nat cnat 16648   FuncCat cfuc 16649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-ixp 7951  df-cat 16376  df-cid 16377  df-homf 16378  df-comf 16379  df-func 16565  df-nat 16650  df-fuc 16651
This theorem is referenced by:  oyoncl  16957
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