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Theorem fucval 18056

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 18042	. . . 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 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → 𝑡 = 𝐶)
5		simprr 785	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → 𝑢 = 𝐷)
6	4, 5	oveq12d 7435	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Func 𝑢) = (𝐶 Func 𝐷))
7		fucval.b	. . . . . 6 ⊢ 𝐵 = (𝐶 Func 𝐷)
8	6, 7	eqtr4di 2815	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Func 𝑢) = 𝐵)
9	8	opeq2d 4843	. . . 4 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ⟨(Base‘ndx), (𝑡 Func 𝑢)⟩ = ⟨(Base‘ndx), 𝐵⟩)
10	4, 5	oveq12d 7435	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = (𝐶 Nat 𝐷))
11		fucval.n	. . . . . 6 ⊢ 𝑁 = (𝐶 Nat 𝐷)
12	10, 11	eqtr4di 2815	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = 𝑁)
13	12	opeq2d 4843	. . . 4 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩ = ⟨(Hom ‘ndx), 𝑁⟩)
14	8	sqxpeqd 5691	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)) = (𝐵 × 𝐵))
15	12	oveqd 7434	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑔(𝑡 Nat 𝑢)ℎ) = (𝑔𝑁ℎ))
16	12	oveqd 7434	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑓(𝑡 Nat 𝑢)𝑔) = (𝑓𝑁𝑔))
17	4	fveq2d 6886	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (Base‘𝑡) = (Base‘𝐶))
18		fucval.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Base‘𝐶)
19	17, 18	eqtr4di 2815	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (Base‘𝑡) = 𝐴)
20	5	fveq2d 6886	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (comp‘𝑢) = (comp‘𝐷))
21		fucval.o	. . . . . . . . . . . . . 14 ⊢ · = (comp‘𝐷)
22	20, 21	eqtr4di 2815	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (comp‘𝑢) = · )
23	22	oveqd 7434	. . . . . . . . . . . 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 7434	. . . . . . . . . . 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 5196	. . . . . . . . . 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 7492	. . . . . . . . 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 3857	. . . . . . . 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 3857	. . . . . . 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 7492	. . . . . 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 486	. . . . . 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 2800	. . . . 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 4843	. . . 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 4712	. . 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 7751	. . . 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 7569	. 2 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ∙ ⟩})
40	1, 39	eqtrid 2809	1 ⊢ (𝜑 → 𝑄 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ∙ ⟩})

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3453 ⦋csb 3850 {ctp 4591 ⟨cop 4593 ↦ cmpt 5190 × cxp 5657 ‘cfv 6537 (class class class)co 7417 ∈ cmpo 7419 1^st c1st 7988 2^nd c2nd 7989 ndxcnx 17291 Basecbs 17307 Hom chom 17359 compcco 17360 Catccat 17758 Func cfunc 17949 Nat cnat 18039 FuncCat cfuc 18040

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-pr 5402 ax-un 7740

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-iota 6493 df-fun 6539 df-fv 6545 df-ov 7420 df-oprab 7421 df-mpo 7422 df-fuc 18042

This theorem is used by: fuccofval 18057 fucbas 18058 fuchom 18059 fucpropd 18075 catcfuccl 18213

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