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Theorem fucval 17806

Description: Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)

**Hypotheses**
Ref	Expression
fucval.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fucval.b	⊢ 𝐵 = (𝐶 Func 𝐷)
fucval.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
fucval.a	⊢ 𝐴 = (Base‘𝐶)
fucval.o	⊢ · = (comp‘𝐷)
fucval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
fucval.d	⊢ (𝜑 → 𝐷 ∈ Cat)
fucval.x	⊢ (𝜑 → ∙ = (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))

**Assertion**
Ref	Expression
fucval	⊢ (𝜑 → 𝑄 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ∙ ⟩})

Distinct variable groups: 𝑣,ℎ,𝐵 𝑎,𝑏,𝑓,𝑔,ℎ,𝑣,𝑥,𝜑 𝐶,𝑎,𝑏,𝑓,𝑔,ℎ,𝑣,𝑥 𝐷,𝑎,𝑏,𝑓,𝑔,ℎ,𝑣,𝑥 Allowed substitution hints: 𝐴(𝑥,𝑣,𝑓,𝑔,ℎ,𝑎,𝑏) 𝐵(𝑥,𝑓,𝑔,𝑎,𝑏) 𝑄(𝑥,𝑣,𝑓,𝑔,ℎ,𝑎,𝑏) ∙ (𝑥,𝑣,𝑓,𝑔,ℎ,𝑎,𝑏) · (𝑥,𝑣,𝑓,𝑔,ℎ,𝑎,𝑏) 𝑁(𝑥,𝑣,𝑓,𝑔,ℎ,𝑎,𝑏)

Proof of Theorem fucval Dummy variables 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucval.q	. 2 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2		df-fuc 17791	. . . 4 ⊢ FuncCat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨(Base‘ndx), (𝑡 Func 𝑢)⟩, ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ℎ ∈ (𝑡 Func 𝑢) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)ℎ), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩})
3	2	a1i 11	. . 3 ⊢ (𝜑 → FuncCat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨(Base‘ndx), (𝑡 Func 𝑢)⟩, ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ℎ ∈ (𝑡 Func 𝑢) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)ℎ), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩}))
4		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → 𝑡 = 𝐶)
5		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → 𝑢 = 𝐷)
6	4, 5	oveq12d 7370	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Func 𝑢) = (𝐶 Func 𝐷))
7		fucval.b	. . . . . 6 ⊢ 𝐵 = (𝐶 Func 𝐷)
8	6, 7	eqtr4di 2796	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Func 𝑢) = 𝐵)
9	8	opeq2d 4836	. . . 4 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ⟨(Base‘ndx), (𝑡 Func 𝑢)⟩ = ⟨(Base‘ndx), 𝐵⟩)
10	4, 5	oveq12d 7370	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = (𝐶 Nat 𝐷))
11		fucval.n	. . . . . 6 ⊢ 𝑁 = (𝐶 Nat 𝐷)
12	10, 11	eqtr4di 2796	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = 𝑁)
13	12	opeq2d 4836	. . . 4 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩ = ⟨(Hom ‘ndx), 𝑁⟩)
14	8	sqxpeqd 5664	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)) = (𝐵 × 𝐵))
15	12	oveqd 7369	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑔(𝑡 Nat 𝑢)ℎ) = (𝑔𝑁ℎ))
16	12	oveqd 7369	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑓(𝑡 Nat 𝑢)𝑔) = (𝑓𝑁𝑔))
17	4	fveq2d 6844	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (Base‘𝑡) = (Base‘𝐶))
18		fucval.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Base‘𝐶)
19	17, 18	eqtr4di 2796	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (Base‘𝑡) = 𝐴)
20	5	fveq2d 6844	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (comp‘𝑢) = (comp‘𝐷))
21		fucval.o	. . . . . . . . . . . . . 14 ⊢ · = (comp‘𝐷)
22	20, 21	eqtr4di 2796	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (comp‘𝑢) = · )
23	22	oveqd 7369	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1^st ‘ℎ)‘𝑥)) = (⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥)))
24	23	oveqd 7369	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)) = ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))
25	19, 24	mpteq12dv 5195	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))
26	15, 16, 25	mpoeq123dv 7427	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑏 ∈ (𝑔(𝑡 Nat 𝑢)ℎ), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = (𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
27	26	csbeq2dv 3861	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)ℎ), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = ⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
28	27	csbeq2dv 3861	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)ℎ), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
29	14, 8, 28	mpoeq123dv 7427	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ℎ ∈ (𝑡 Func 𝑢) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)ℎ), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) = (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))
30		fucval.x	. . . . . . 7 ⊢ (𝜑 → ∙ = (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))
31	30	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ∙ = (𝑣 ∈ (𝐵 × 𝐵), ℎ ∈ 𝐵 ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))
32	29, 31	eqtr4d 2781	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ℎ ∈ (𝑡 Func 𝑢) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)ℎ), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) = ∙ )
33	32	opeq2d 4836	. . . 4 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ⟨(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ℎ ∈ (𝑡 Func 𝑢) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)ℎ), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩ = ⟨(comp‘ndx), ∙ ⟩)
34	9, 13, 33	tpeq123d 4708	. . 3 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → {⟨(Base‘ndx), (𝑡 Func 𝑢)⟩, ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ℎ ∈ (𝑡 Func 𝑢) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)ℎ), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑢)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ∙ ⟩})
35		fucval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
36		fucval.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
37		tpex 7674	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ∙ ⟩} ∈ V
38	37	a1i 11	. . 3 ⊢ (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ∙ ⟩} ∈ V)
39	3, 34, 35, 36, 38	ovmpod 7502	. 2 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ∙ ⟩})
40	1, 39	eqtrid 2790	1 ⊢ (𝜑 → 𝑄 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ∙ ⟩})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ⦋csb 3854 {ctp 4589 ⟨cop 4591 ↦ cmpt 5187 × cxp 5630 ‘cfv 6494 (class class class)co 7352 ∈ cmpo 7354 1^st c1st 7912 2^nd c2nd 7913 ndxcnx 17025 Basecbs 17043 Hom chom 17104 compcco 17105 Catccat 17504 Func cfunc 17700 Nat cnat 17788 FuncCat cfuc 17789

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7665

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-iota 6446 df-fun 6496 df-fv 6502 df-ov 7355 df-oprab 7356 df-mpo 7357 df-fuc 17791

This theorem is referenced by: fuccofval 17807 fucbas 17808 fuchom 17809 fuchomOLD 17810 fucpropd 17826 catcfuccl 17965 catcfucclOLD 17966

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