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Theorem fucpropd 16406
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 16329 . . . 4 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
109opeq2d 4341 . . 3 (𝜑 → ⟨(Base‘ndx), (𝐴 Func 𝐶)⟩ = ⟨(Base‘ndx), (𝐵 Func 𝐷)⟩)
111, 2, 3, 4, 5, 6, 7, 8natpropd 16405 . . . 4 (𝜑 → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
1211opeq2d 4341 . . 3 (𝜑 → ⟨(Hom ‘ndx), (𝐴 Nat 𝐶)⟩ = ⟨(Hom ‘ndx), (𝐵 Nat 𝐷)⟩)
139sqxpeqd 5055 . . . . 5 (𝜑 → ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) = ((𝐵 Func 𝐷) × (𝐵 Func 𝐷)))
149adantr 479 . . . . 5 ((𝜑𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶))) → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
15 nfv 1829 . . . . . 6 𝑓(𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶)))
16 nfcsb1v 3514 . . . . . . 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 6098 . . . . . . 7 (1st𝑣) ∈ V
1918a1i 11 . . . . . 6 ((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) → (1st𝑣) ∈ V)
20 nfv 1829 . . . . . . . 8 𝑔((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣))
21 nfcsb1v 3514 . . . . . . . . 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 6098 . . . . . . . . 9 (2nd𝑣) ∈ V
2423a1i 11 . . . . . . . 8 (((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) → (2nd𝑣) ∈ V)
2511ad3antrrr 761 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑣 ∈ ((𝐴 Func 𝐶) × (𝐴 Func 𝐶)) ∧ ∈ (𝐴 Func 𝐶))) ∧ 𝑓 = (1st𝑣)) ∧ 𝑔 = (2nd𝑣)) → (𝐴 Nat 𝐶) = (𝐵 Nat 𝐷))
2625oveqd 6544 . . . . . . . . . 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 6551 . . . . . . . . . 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 16125 . . . . . . . . . . . 12 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
2928ad4antr 763 . . . . . . . . . . 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 2609 . . . . . . . . . . . 12 (Base‘𝐶) = (Base‘𝐶)
31 eqid 2609 . . . . . . . . . . . 12 (Hom ‘𝐶) = (Hom ‘𝐶)
32 eqid 2609 . . . . . . . . . . . 12 (comp‘𝐶) = (comp‘𝐶)
33 eqid 2609 . . . . . . . . . . . 12 (comp‘𝐷) = (comp‘𝐷)
343ad5antr 765 . . . . . . . . . . . 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 765 . . . . . . . . . . . 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 2609 . . . . . . . . . . . . . 14 (Base‘𝐴) = (Base‘𝐴)
37 relfunc 16291 . . . . . . . . . . . . . . 15 Rel (𝐴 Func 𝐶)
38 simpllr 794 . . . . . . . . . . . . . . . 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 802 . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . 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 7066 . . . . . . . . . . . . . . . . 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 2687 . . . . . . . . . . . . . . 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 7085 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑓 ∈ (𝐴 Func 𝐶)) → (1st𝑓)(𝐴 Func 𝐶)(2nd𝑓))
4537, 43, 44sylancr 693 . . . . . . . . . . . . . 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 16295 . . . . . . . . . . . . 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 6252 . . . . . . . . . . . 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 787 . . . . . . . . . . . . . . . 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 7067 . . . . . . . . . . . . . . . . 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 2687 . . . . . . . . . . . . . . 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 7085 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ 𝑔 ∈ (𝐴 Func 𝐶)) → (1st𝑔)(𝐴 Func 𝐶)(2nd𝑔))
5337, 51, 52sylancr 693 . . . . . . . . . . . . . 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 16295 . . . . . . . . . . . . 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 6252 . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . 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 7085 . . . . . . . . . . . . . . 15 ((Rel (𝐴 Func 𝐶) ∧ ∈ (𝐴 Func 𝐶)) → (1st)(𝐴 Func 𝐶)(2nd))
5837, 56, 57sylancr 693 . . . . . . . . . . . . . 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 16295 . . . . . . . . . . . . 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 6252 . . . . . . . . . . . 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 2609 . . . . . . . . . . . . 13 (𝐴 Nat 𝐶) = (𝐴 Nat 𝐶)
62 simplrr 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)) → 𝑎 ∈ (𝑓(𝐴 Nat 𝐶)𝑔))
6361, 62nat1st2nd 16380 . . . . . . . . . . . . 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 475 . . . . . . . . . . . . 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 16382 . . . . . . . . . . . 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 795 . . . . . . . . . . . . . 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 16380 . . . . . . . . . . . . 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 16382 . . . . . . . . . . . 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 16137 . . . . . . . . . . 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 4656 . . . . . . . . . 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 6592 . . . . . . . . 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 3507 . . . . . . . . . 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 480 . . . . . . . . 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 2643 . . . . . . . 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 3519 . . . . . . 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 3507 . . . . . . . 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 480 . . . . . . 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 2643 . . . . . 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 3519 . . . . 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 6592 . . . 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 4341 . . 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 4226 . 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 2609 . . 3 (𝐴 FuncCat 𝐶) = (𝐴 FuncCat 𝐶)
84 eqid 2609 . . 3 (𝐴 Func 𝐶) = (𝐴 Func 𝐶)
85 eqidd 2610 . . 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 16387 . 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 2609 . . 3 (𝐵 FuncCat 𝐷) = (𝐵 FuncCat 𝐷)
88 eqid 2609 . . 3 (𝐵 Func 𝐷) = (𝐵 Func 𝐷)
89 eqid 2609 . . 3 (𝐵 Nat 𝐷) = (𝐵 Nat 𝐷)
90 eqid 2609 . . 3 (Base‘𝐵) = (Base‘𝐵)
91 eqidd 2610 . . 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 16387 . 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 2653 1 (𝜑 → (𝐴 FuncCat 𝐶) = (𝐵 FuncCat 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wnfc 2737  Vcvv 3172  csb 3498  {ctp 4128  cop 4130   class class class wbr 4577  cmpt 4637   × cxp 5026  Rel wrel 5033  cfv 5790  (class class class)co 6527  cmpt2 6529  1st c1st 7034  2nd c2nd 7035  ndxcnx 15638  Basecbs 15641  Hom chom 15725  compcco 15726  Catccat 16094  Homf chomf 16096  compfccomf 16097   Func cfunc 16283   Nat cnat 16370   FuncCat cfuc 16371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-map 7723  df-ixp 7772  df-cat 16098  df-cid 16099  df-homf 16100  df-comf 16101  df-func 16287  df-nat 16372  df-fuc 16373
This theorem is referenced by:  oyoncl  16679
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