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Theorem natfval 16653
Description: Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
natfval.1 𝑁 = (𝐶 Nat 𝐷)
natfval.b 𝐵 = (Base‘𝐶)
natfval.h 𝐻 = (Hom ‘𝐶)
natfval.j 𝐽 = (Hom ‘𝐷)
natfval.o · = (comp‘𝐷)
Assertion
Ref Expression
natfval 𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))})
Distinct variable groups:   𝑓,𝑎,𝑔,,𝑟,𝑠,𝑥,𝑦   𝐵,𝑎,𝑓,𝑔,𝑟,𝑠,𝑥,𝑦   𝐶,𝑎,𝑓,𝑔,,𝑟,𝑠,𝑥,𝑦   𝐽,𝑎,𝑓,𝑔,𝑟,𝑠   𝐻,𝑎,𝑓,𝑔,,𝑟,𝑠   · ,𝑎,𝑓,𝑔,𝑟,𝑠   𝐷,𝑎,𝑓,𝑔,,𝑟,𝑠,𝑥,𝑦
Allowed substitution hints:   𝐵()   · (𝑥,𝑦,)   𝐻(𝑥,𝑦)   𝐽(𝑥,𝑦,)   𝑁(𝑥,𝑦,𝑓,𝑔,,𝑠,𝑟,𝑎)

Proof of Theorem natfval
Dummy variables 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 natfval.1 . 2 𝑁 = (𝐶 Nat 𝐷)
2 oveq12 6699 . . . . 5 ((𝑡 = 𝐶𝑢 = 𝐷) → (𝑡 Func 𝑢) = (𝐶 Func 𝐷))
3 simpl 472 . . . . . . . . . . . 12 ((𝑡 = 𝐶𝑢 = 𝐷) → 𝑡 = 𝐶)
43fveq2d 6233 . . . . . . . . . . 11 ((𝑡 = 𝐶𝑢 = 𝐷) → (Base‘𝑡) = (Base‘𝐶))
5 natfval.b . . . . . . . . . . 11 𝐵 = (Base‘𝐶)
64, 5syl6eqr 2703 . . . . . . . . . 10 ((𝑡 = 𝐶𝑢 = 𝐷) → (Base‘𝑡) = 𝐵)
76ixpeq1d 7962 . . . . . . . . 9 ((𝑡 = 𝐶𝑢 = 𝐷) → X𝑥 ∈ (Base‘𝑡)((𝑟𝑥)(Hom ‘𝑢)(𝑠𝑥)) = X𝑥𝐵 ((𝑟𝑥)(Hom ‘𝑢)(𝑠𝑥)))
8 simpr 476 . . . . . . . . . . . . 13 ((𝑡 = 𝐶𝑢 = 𝐷) → 𝑢 = 𝐷)
98fveq2d 6233 . . . . . . . . . . . 12 ((𝑡 = 𝐶𝑢 = 𝐷) → (Hom ‘𝑢) = (Hom ‘𝐷))
10 natfval.j . . . . . . . . . . . 12 𝐽 = (Hom ‘𝐷)
119, 10syl6eqr 2703 . . . . . . . . . . 11 ((𝑡 = 𝐶𝑢 = 𝐷) → (Hom ‘𝑢) = 𝐽)
1211oveqd 6707 . . . . . . . . . 10 ((𝑡 = 𝐶𝑢 = 𝐷) → ((𝑟𝑥)(Hom ‘𝑢)(𝑠𝑥)) = ((𝑟𝑥)𝐽(𝑠𝑥)))
1312ixpeq2dv 7966 . . . . . . . . 9 ((𝑡 = 𝐶𝑢 = 𝐷) → X𝑥𝐵 ((𝑟𝑥)(Hom ‘𝑢)(𝑠𝑥)) = X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)))
147, 13eqtrd 2685 . . . . . . . 8 ((𝑡 = 𝐶𝑢 = 𝐷) → X𝑥 ∈ (Base‘𝑡)((𝑟𝑥)(Hom ‘𝑢)(𝑠𝑥)) = X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)))
153fveq2d 6233 . . . . . . . . . . . . 13 ((𝑡 = 𝐶𝑢 = 𝐷) → (Hom ‘𝑡) = (Hom ‘𝐶))
16 natfval.h . . . . . . . . . . . . 13 𝐻 = (Hom ‘𝐶)
1715, 16syl6eqr 2703 . . . . . . . . . . . 12 ((𝑡 = 𝐶𝑢 = 𝐷) → (Hom ‘𝑡) = 𝐻)
1817oveqd 6707 . . . . . . . . . . 11 ((𝑡 = 𝐶𝑢 = 𝐷) → (𝑥(Hom ‘𝑡)𝑦) = (𝑥𝐻𝑦))
198fveq2d 6233 . . . . . . . . . . . . . . 15 ((𝑡 = 𝐶𝑢 = 𝐷) → (comp‘𝑢) = (comp‘𝐷))
20 natfval.o . . . . . . . . . . . . . . 15 · = (comp‘𝐷)
2119, 20syl6eqr 2703 . . . . . . . . . . . . . 14 ((𝑡 = 𝐶𝑢 = 𝐷) → (comp‘𝑢) = · )
2221oveqd 6707 . . . . . . . . . . . . 13 ((𝑡 = 𝐶𝑢 = 𝐷) → (⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦)) = (⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦)))
2322oveqd 6707 . . . . . . . . . . . 12 ((𝑡 = 𝐶𝑢 = 𝐷) → ((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = ((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)))
2421oveqd 6707 . . . . . . . . . . . . 13 ((𝑡 = 𝐶𝑢 = 𝐷) → (⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦)) = (⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦)))
2524oveqd 6707 . . . . . . . . . . . 12 ((𝑡 = 𝐶𝑢 = 𝐷) → (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦))(𝑎𝑥)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥)))
2623, 25eqeq12d 2666 . . . . . . . . . . 11 ((𝑡 = 𝐶𝑢 = 𝐷) → (((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦))(𝑎𝑥)) ↔ ((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))))
2718, 26raleqbidv 3182 . . . . . . . . . 10 ((𝑡 = 𝐶𝑢 = 𝐷) → (∀ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦))(𝑎𝑥)) ↔ ∀ ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))))
286, 27raleqbidv 3182 . . . . . . . . 9 ((𝑡 = 𝐶𝑢 = 𝐷) → (∀𝑦 ∈ (Base‘𝑡)∀ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦))(𝑎𝑥)) ↔ ∀𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))))
296, 28raleqbidv 3182 . . . . . . . 8 ((𝑡 = 𝐶𝑢 = 𝐷) → (∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦))(𝑎𝑥)) ↔ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))))
3014, 29rabeqbidv 3226 . . . . . . 7 ((𝑡 = 𝐶𝑢 = 𝐷) → {𝑎X𝑥 ∈ (Base‘𝑡)((𝑟𝑥)(Hom ‘𝑢)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦))(𝑎𝑥))} = {𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))})
3130csbeq2dv 4025 . . . . . 6 ((𝑡 = 𝐶𝑢 = 𝐷) → (1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝑡)((𝑟𝑥)(Hom ‘𝑢)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦))(𝑎𝑥))} = (1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))})
3231csbeq2dv 4025 . . . . 5 ((𝑡 = 𝐶𝑢 = 𝐷) → (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝑡)((𝑟𝑥)(Hom ‘𝑢)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦))(𝑎𝑥))} = (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))})
332, 2, 32mpt2eq123dv 6759 . . . 4 ((𝑡 = 𝐶𝑢 = 𝐷) → (𝑓 ∈ (𝑡 Func 𝑢), 𝑔 ∈ (𝑡 Func 𝑢) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝑡)((𝑟𝑥)(Hom ‘𝑢)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦))(𝑎𝑥))}) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))}))
34 df-nat 16650 . . . 4 Nat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ (𝑓 ∈ (𝑡 Func 𝑢), 𝑔 ∈ (𝑡 Func 𝑢) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥 ∈ (Base‘𝑡)((𝑟𝑥)(Hom ‘𝑢)(𝑠𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩(comp‘𝑢)(𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩(comp‘𝑢)(𝑠𝑦))(𝑎𝑥))}))
35 ovex 6718 . . . . 5 (𝐶 Func 𝐷) ∈ V
3635, 35mpt2ex 7292 . . . 4 (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))}) ∈ V
3733, 34, 36ovmpt2a 6833 . . 3 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Nat 𝐷) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))}))
3834mpt2ndm0 6917 . . . 4 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Nat 𝐷) = ∅)
39 funcrcl 16570 . . . . . . . 8 (𝑓 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
4039con3i 150 . . . . . . 7 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ¬ 𝑓 ∈ (𝐶 Func 𝐷))
4140eq0rdv 4012 . . . . . 6 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = ∅)
42 mpt2eq12 6757 . . . . . 6 (((𝐶 Func 𝐷) = ∅ ∧ (𝐶 Func 𝐷) = ∅) → (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))}) = (𝑓 ∈ ∅, 𝑔 ∈ ∅ ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))}))
4341, 41, 42syl2anc 694 . . . . 5 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))}) = (𝑓 ∈ ∅, 𝑔 ∈ ∅ ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))}))
44 mpt20 6767 . . . . 5 (𝑓 ∈ ∅, 𝑔 ∈ ∅ ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))}) = ∅
4543, 44syl6eq 2701 . . . 4 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))}) = ∅)
4638, 45eqtr4d 2688 . . 3 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Nat 𝐷) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))}))
4737, 46pm2.61i 176 . 2 (𝐶 Nat 𝐷) = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))})
481, 47eqtri 2673 1 𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  csb 3566  c0 3948  cop 4216  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  Xcixp 7950  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372   Func cfunc 16561   Nat cnat 16648
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-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-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-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-ixp 7951  df-func 16565  df-nat 16650
This theorem is referenced by:  isnat  16654  natffn  16656  natrcl  16657  wunnat  16663  natpropd  16683
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