fucval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > fucval		Structured version Visualization version GIF version

Theorem fucval 18008

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 17994	. . . 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 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → 𝑡 = 𝐶)
5		simprr 784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → 𝑢 = 𝐷)
6	4, 5	oveq12d 7418	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Func 𝑢) = (𝐶 Func 𝐷))
7		fucval.b	. . . . . 6 ⊢ 𝐵 = (𝐶 Func 𝐷)
8	6, 7	eqtr4di 2818	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Func 𝑢) = 𝐵)
9	8	opeq2d 4841	. . . 4 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ⟨(Base‘ndx), (𝑡 Func 𝑢)⟩ = ⟨(Base‘ndx), 𝐵⟩)
10	4, 5	oveq12d 7418	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = (𝐶 Nat 𝐷))
11		fucval.n	. . . . . 6 ⊢ 𝑁 = (𝐶 Nat 𝐷)
12	10, 11	eqtr4di 2818	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = 𝑁)
13	12	opeq2d 4841	. . . 4 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩ = ⟨(Hom ‘ndx), 𝑁⟩)
14	8	sqxpeqd 5684	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)) = (𝐵 × 𝐵))
15	12	oveqd 7417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑔(𝑡 Nat 𝑢)ℎ) = (𝑔𝑁ℎ))
16	12	oveqd 7417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝑓(𝑡 Nat 𝑢)𝑔) = (𝑓𝑁𝑔))
17	4	fveq2d 6875	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (Base‘𝑡) = (Base‘𝐶))
18		fucval.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Base‘𝐶)
19	17, 18	eqtr4di 2818	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (Base‘𝑡) = 𝐴)
20	5	fveq2d 6875	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (comp‘𝑢) = (comp‘𝐷))
21		fucval.o	. . . . . . . . . . . . . 14 ⊢ · = (comp‘𝐷)
22	20, 21	eqtr4di 2818	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 = 𝐶 ∧ 𝑢 = 𝐷)) → (comp‘𝑢) = · )
23	22	oveqd 7417	. . . . . . . . . . . 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 7417	. . . . . . . . . . 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 5192	. . . . . . . . . 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 7475	. . . . . . . . 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 3862	. . . . . . . 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 3862	. . . . . . 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 7475	. . . . . 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 485	. . . . . 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 2803	. . . . 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 4841	. . . 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 4710	. . 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 7733	. . . 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 7552	. 2 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ∙ ⟩})
40	1, 39	eqtrid 2812	1 ⊢ (𝜑 → 𝑄 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ∙ ⟩})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 Vcvv 3457 ⦋csb 3855 {ctp 4589 ⟨cop 4591 ↦ cmpt 5186 × cxp 5650 ‘cfv 6525 (class class class)co 7400 ∈ cmpo 7402 1^st c1st 7972 2^nd c2nd 7973 ndxcnx 17243 Basecbs 17259 Hom chom 17311 compcco 17312 Catccat 17710 Func cfunc 17901 Nat cnat 17991 FuncCat cfuc 17992

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-pr 5395 ax-un 7722

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-br 5106 df-opab 5168 df-mpt 5187 df-id 5547 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-iota 6481 df-fun 6527 df-fv 6533 df-ov 7403 df-oprab 7404 df-mpo 7405 df-fuc 17994

This theorem is referenced by: fuccofval 18009 fucbas 18010 fuchom 18011 fucpropd 18027 catcfuccl 18165

W3C validator