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Theorem fucval 17865

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 17851	. . . 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 7364	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Func 𝑢) = (𝐶 Func 𝐷))
7		fucval.b	. . . . . 6 ⊢ 𝐵 = (𝐶 Func 𝐷)
8	6, 7	eqtr4di 2784	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Func 𝑢) = 𝐵)
9	8	opeq2d 4832	. . . 4 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ⟨(Base‘ndx), (𝑡 Func 𝑢)⟩ = ⟨(Base‘ndx), 𝐵⟩)
10	4, 5	oveq12d 7364	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = (𝐶 Nat 𝐷))
11		fucval.n	. . . . . 6 ⊢ 𝑁 = (𝐶 Nat 𝐷)
12	10, 11	eqtr4di 2784	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = 𝑁)
13	12	opeq2d 4832	. . . 4 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩ = ⟨(Hom ‘ndx), 𝑁⟩)
14	8	sqxpeqd 5648	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)) = (𝐵 × 𝐵))
15	12	oveqd 7363	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑔(𝑡 Nat 𝑢)ℎ) = (𝑔𝑁ℎ))
16	12	oveqd 7363	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑓(𝑡 Nat 𝑢)𝑔) = (𝑓𝑁𝑔))
17	4	fveq2d 6826	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (Base‘𝑡) = (Base‘𝐶))
18		fucval.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Base‘𝐶)
19	17, 18	eqtr4di 2784	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (Base‘𝑡) = 𝐴)
20	5	fveq2d 6826	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (comp‘𝑢) = (comp‘𝐷))
21		fucval.o	. . . . . . . . . . . . . 14 ⊢ · = (comp‘𝐷)
22	20, 21	eqtr4di 2784	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (comp‘𝑢) = · )
23	22	oveqd 7363	. . . . . . . . . . . 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 7363	. . . . . . . . . . 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 5178	. . . . . . . . . 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 7421	. . . . . . . . 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 7421	. . . . . 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 480	. . . . . 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 2769	. . . . 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 4832	. . . 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 4701	. . 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 7679	. . . 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 7498	. 2 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ∙ ⟩})
40	1, 39	eqtrid 2778	1 ⊢ (𝜑 → 𝑄 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ∙ ⟩})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 Vcvv 3436 ⦋csb 3850 {ctp 4580 ⟨cop 4582 ↦ cmpt 5172 × cxp 5614 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 1^st c1st 7919 2^nd c2nd 7920 ndxcnx 17101 Basecbs 17117 Hom chom 17169 compcco 17170 Catccat 17567 Func cfunc 17758 Nat cnat 17848 FuncCat cfuc 17849

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pr 5370 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-ss 3919 df-nul 4284 df-if 4476 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-iota 6437 df-fun 6483 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-fuc 17851

This theorem is referenced by: fuccofval 17866 fucbas 17867 fuchom 17868 fucpropd 17884 catcfuccl 18022

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