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Theorem fucpropd 16460
 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 16383 . . . 4 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
109opeq2d 4347 . . 3 (𝜑 → ⟨(Base‘ndx), (𝐴 Func 𝐶)⟩ = ⟨(Base‘ndx), (𝐵 Func 𝐷)⟩)
111, 2, 3, 4, 5, 6, 7, 8natpropd 16459 . . . 4 (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
1211opeq2d 4347 . . 3 (𝜑 → ⟨(Hom ‘ndx), (𝐴 Nat 𝐶)⟩ = ⟨(Hom ‘ndx), (𝐵 Nat 𝐷)⟩)
139sqxpeqd 5065 . . . . 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 1830 . . . . . 6 𝑓(𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶)))
16 nfcsb1v 3515 . . . . . . 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 fvex 6113 . . . . . . 7 (1st𝑣) ∈ V
1918a1i 11 . . . . . 6 ((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) → (1st𝑣) ∈ V)
20 nfv 1830 . . . . . . . 8 𝑔((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣))
21 nfcsb1v 3515 . . . . . . . . 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))))
2221a1i 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)))))
23 fvex 6113 . . . . . . . . 9 (2nd𝑣) ∈ V
2423a1i 11 . . . . . . . 8 (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) → (2nd𝑣) ∈ V)
2511ad3antrrr 762 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
2625oveqd 6566 . . . . . . . . . 10 ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) → (𝑔(𝐴 Nat 𝐶)) = (𝑔(𝐵 Nat 𝐷)))
2725oveqdr 6573 . . . . . . . . . 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))
281homfeqbas 16179 . . . . . . . . . . . 12 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
2928ad4antr 764 . . . . . . . . . . 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))
30 eqid 2610 . . . . . . . . . . . 12 (Base‘𝐶) = (Base‘𝐶)
31 eqid 2610 . . . . . . . . . . . 12 (Hom ‘𝐶) = (Hom ‘𝐶)
32 eqid 2610 . . . . . . . . . . . 12 (comp‘𝐶) = (comp‘𝐶)
33 eqid 2610 . . . . . . . . . . . 12 (comp‘𝐷) = (comp‘𝐷)
343ad5antr 766 . . . . . . . . . . . 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))
354ad5antr 766 . . . . . . . . . . . 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))
36 eqid 2610 . . . . . . . . . . . . . 14 (Base‘𝐴) = (Base‘𝐴)
37 relfunc 16345 . . . . . . . . . . . . . . 15 Rel (𝐴 Func 𝐶)
38 simpllr 795 . . . . . . . . . . . . . . . 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))
39 simp-4r 803 . . . . . . . . . . . . . . . . . 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)))
4039simpld 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)))
41 xp1st 7089 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) → (1st𝑣) ∈ (𝐴 Func 𝐶))
4240, 41syl 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))
4338, 42eqeltrd 2688 . . . . . . . . . . . . . . 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))
44 1st2ndbr 7108 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
4537, 43, 44sylancr 694 . . . . . . . . . . . . . 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))
4636, 30, 45funcf1 16349 . . . . . . . . . . . . 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))
4746ffvelrnda 6267 . . . . . . . . . . . 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))
48 simplr 788 . . . . . . . . . . . . . . . 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))
49 xp2nd 7090 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) → (2nd𝑣) ∈ (𝐴 Func 𝐶))
5040, 49syl 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))
5148, 50eqeltrd 2688 . . . . . . . . . . . . . . 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))
52 1st2ndbr 7108 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
5337, 51, 52sylancr 694 . . . . . . . . . . . . . 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))
5436, 30, 53funcf1 16349 . . . . . . . . . . . . 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))
5554ffvelrnda 6267 . . . . . . . . . . . 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))
5639simprd 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))
57 1st2ndbr 7108 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ ∈ (𝐴 Func 𝐶)) → (1st)(𝐴 Func 𝐶)(2nd))
5837, 56, 57sylancr 694 . . . . . . . . . . . . . 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))
5936, 30, 58funcf1 16349 . . . . . . . . . . . . 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))
6059ffvelrnda 6267 . . . . . . . . . . . 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))
61 eqid 2610 . . . . . . . . . . . . 13 (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
62 simplrr 797 . . . . . . . . . . . . . 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))
6361, 62nat1st2nd 16434 . . . . . . . . . . . . 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)⟩))
64 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))
6561, 63, 36, 31, 64natcl 16436 . . . . . . . . . . . 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)))
66 simplrl 796 . . . . . . . . . . . . . 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)))
6761, 66nat1st2nd 16434 . . . . . . . . . . . . 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)⟩))
6861, 67, 36, 31, 64natcl 16436 . . . . . . . . . . . 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)))
6930, 31, 32, 33, 34, 35, 47, 55, 60, 65, 68comfeqval 16191 . . . . . . . . . . 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)))
7029, 69mpteq12dva 4662 . . . . . . . . . 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))))
7126, 27, 70mpt2eq123dva 6614 . . . . . . . . 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)))))
72 csbeq1a 3508 . . . . . . . . . 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)))))
7372adantl 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)))))
7471, 73eqtrd 2644 . . . . . . . 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)))))
7520, 22, 24, 74csbiedf 3520 . . . . . . 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)))))
76 csbeq1a 3508 . . . . . . . 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)))))
7776adantl 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)))))
7875, 77eqtrd 2644 . . . . . 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)))))
7915, 17, 19, 78csbiedf 3520 . . . . 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)))))
8013, 14, 79mpt2eq123dva 6614 . . . 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))))))
8180opeq2d 4347 . . 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)))))⟩)
8210, 12, 81tpeq123d 4227 . 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)))))⟩})
83 eqid 2610 . . 3 (𝐴 FuncCat 𝐶) = (𝐴 FuncCat 𝐶)
84 eqid 2610 . . 3 (𝐴 Func 𝐶) = (𝐴 Func 𝐶)
85 eqidd 2611 . . 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))))))
8683, 84, 61, 36, 32, 5, 7, 85fucval 16441 . 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)))))⟩})
87 eqid 2610 . . 3 (𝐵 FuncCat 𝐷) = (𝐵 FuncCat 𝐷)
88 eqid 2610 . . 3 (𝐵 Func 𝐷) = (𝐵 Func 𝐷)
89 eqid 2610 . . 3 (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
90 eqid 2610 . . 3 (Base‘𝐵) = (Base‘𝐵)
91 eqidd 2611 . . 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))))))
9287, 88, 89, 90, 33, 6, 8, 91fucval 16441 . 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)))))⟩})
9382, 86, 923eqtr4d 2654 1 (𝜑 → (𝐴 FuncCat 𝐶) = (𝐵 FuncCat 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  Vcvv 3173  ⦋csb 3499  {ctp 4129  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  ndxcnx 15692  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Homf chomf 16150  compfccomf 16151   Func cfunc 16337   Nat cnat 16424   FuncCat cfuc 16425 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ixp 7795  df-cat 16152  df-cid 16153  df-homf 16154  df-comf 16155  df-func 16341  df-nat 16426  df-fuc 16427 This theorem is referenced by:  oyoncl  16733
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