Theorem lmddu 50445

Description: The duality of limits and colimits: limits of a diagram are colimits of an opposite diagram in opposite categories. (Contributed by Zhi Wang, 20-Nov-2025.)

Hypotheses

Ref	Expression
lmddu.o	⊢ 𝑂 = (oppCat‘𝐶)
lmddu.p	⊢ 𝑃 = (oppCat‘𝐷)
lmddu.g	⊢ 𝐺 = ( oppFunc ‘𝐹)
lmddu.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
lmddu.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
lmddu	⊢ (𝜑 → ((𝐶 Limit 𝐷)‘𝐹) = ((𝑂 Colimit 𝑃)‘𝐺))

Proof of Theorem lmddu

Dummy variables 𝑓 𝑔 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmddu.o	. . . . 5 ⊢ 𝑂 = (oppCat‘𝐶)
2	1	oveq1i 7420	. . . 4 ⊢ (𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶))) = ((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))
3	2	oveqi 7423	. . 3 ⊢ (( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
4		relup 49961	. . . 4 ⊢ Rel (( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
5		relup 49961	. . . 4 ⊢ Rel ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)
6		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚) → 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚)
7		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚)
8		lmddu.p	. . . . . . . 8 ⊢ 𝑃 = (oppCat‘𝐷)
9		lmddu.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
10	9	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐷 ∈ 𝑊)
11		lmddu.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
12	11	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐶 ∈ 𝑉)
13		lmddu.g	. . . . . . . . 9 ⊢ 𝐺 = ( oppFunc ‘𝐹)
14	7	up1st2nd 49963	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑥(⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚)
15		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑃 FuncCat 𝑂) = (𝑃 FuncCat 𝑂)
16	15	fucbas 18015	. . . . . . . . . 10 ⊢ (𝑃 Func 𝑂) = (Base‘(𝑃 FuncCat 𝑂))
17	14, 16	uprcl3 49968	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐺 ∈ (𝑃 Func 𝑂))
18	13, 17	eqeltrrid 2868	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → ( oppFunc ‘𝐹) ∈ (𝑃 Func 𝑂))
19	8, 1, 10, 12, 18	funcoppc5 49923	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐹 ∈ (𝐷 Func 𝐶))
20		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
21	14, 1, 20	oppcuprcl4 49977	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑥 ∈ (Base‘𝐶))
22		eqid 2763	. . . . . . . . . 10 ⊢ (𝑃 Nat 𝑂) = (𝑃 Nat 𝑂)
23	15, 22	fuchom 18016	. . . . . . . . 9 ⊢ (𝑃 Nat 𝑂) = (Hom ‘(𝑃 FuncCat 𝑂))
24	14, 23	uprcl5 49970	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑚 ∈ (𝐺(𝑃 Nat 𝑂)((1st ‘(𝑂Δfunc𝑃))‘𝑥)))
25		eqid 2763	. . . . . . . . 9 ⊢ (𝐷 Nat 𝐶) = (𝐷 Nat 𝐶)
26		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (𝐶Δfunc𝐷) = (𝐶Δfunc𝐷)
27		funcrcl 17915	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → (𝐷 ∈ Cat ∧ 𝐶 ∈ Cat))
28	27	simprd 500	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → 𝐶 ∈ Cat)
29	27	simpld 499	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → 𝐷 ∈ Cat)
30		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
31	26, 28, 29, 30	diagcl 18292	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → (𝐶Δfunc𝐷) ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
32	31	oppf1 49917	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → (1st ‘( oppFunc ‘(𝐶Δfunc𝐷))) = (1st ‘(𝐶Δfunc𝐷)))
33	32	fveq1d 6883	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → ((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥) = ((1st ‘(𝐶Δfunc𝐷))‘𝑥))
34	33	fveq2d 6885	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → ( oppFunc ‘((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)) = ( oppFunc ‘((1st ‘(𝐶Δfunc𝐷))‘𝑥)))
35	19, 34	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → ( oppFunc ‘((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)) = ( oppFunc ‘((1st ‘(𝐶Δfunc𝐷))‘𝑥)))
36	19, 28	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐶 ∈ Cat)
37	19, 29	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐷 ∈ Cat)
38	1, 8, 26, 36, 37, 20, 21	oppfdiag1a 50193	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → ( oppFunc ‘((1st ‘(𝐶Δfunc𝐷))‘𝑥)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑥))
39	35, 38	eqtr2d 2799	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → ((1st ‘(𝑂Δfunc𝑃))‘𝑥) = ( oppFunc ‘((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)))
40	13	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐺 = ( oppFunc ‘𝐹))
41	8, 1, 25, 22, 39, 40, 10, 12	natoppfb 50009	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹) = (𝐺(𝑃 Nat 𝑂)((1st ‘(𝑂Δfunc𝑃))‘𝑥)))
42	24, 41	eleqtrrd 2866	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹))
43		simp1 1154	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝐹 ∈ (𝐷 Func 𝐶))
44	43	fvresd 6901	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → (( oppFunc ↾ (𝐷 Func 𝐶))‘𝐹) = ( oppFunc ‘𝐹))
45	44, 13	eqtr4di 2816	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → (( oppFunc ↾ (𝐷 Func 𝐶))‘𝐹) = 𝐺)
46		eqid 2763	. . . . . . . . . 10 ⊢ (oppCat‘(𝐷 FuncCat 𝐶)) = (oppCat‘(𝐷 FuncCat 𝐶))
47		eqidd 2764	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → ( oppFunc ↾ (𝐷 Func 𝐶)) = ( oppFunc ↾ (𝐷 Func 𝐶)))
48		eqidd 2764	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → (𝑓 ∈ (𝐷 Func 𝐶), 𝑔 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑔(𝐷 Nat 𝐶)𝑓))) = (𝑓 ∈ (𝐷 Func 𝐶), 𝑔 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑔(𝐷 Nat 𝐶)𝑓))))
49	29	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝐷 ∈ Cat)
50	28	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝐶 ∈ Cat)
51	8, 1, 30, 46, 15, 25, 47, 48, 49, 50	fucoppcffth 50189	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → ( oppFunc ↾ (𝐷 Func 𝐶))(((oppCat‘(𝐷 FuncCat 𝐶)) Full (𝑃 FuncCat 𝑂)) ∩ ((oppCat‘(𝐷 FuncCat 𝐶)) Faith (𝑃 FuncCat 𝑂)))(𝑓 ∈ (𝐷 Func 𝐶), 𝑔 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑔(𝐷 Nat 𝐶)𝑓))))
52	1, 8, 26, 50, 49, 47, 25, 48	oppfdiag 50194	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → (⟨( oppFunc ↾ (𝐷 Func 𝐶)), (𝑓 ∈ (𝐷 Func 𝐶), 𝑔 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑔(𝐷 Nat 𝐶)𝑓)))⟩ ∘func ( oppFunc ‘(𝐶Δfunc𝐷))) = (𝑂Δfunc𝑃))
53		relfunc 17914	. . . . . . . . . . . 12 ⊢ Rel (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶)))
54	1, 46, 31	oppfoppc2 49920	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → ( oppFunc ‘(𝐶Δfunc𝐷)) ∈ (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶))))
55	43, 54	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → ( oppFunc ‘(𝐶Δfunc𝐷)) ∈ (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶))))
56		1st2nd 8032	. . . . . . . . . . . 12 ⊢ ((Rel (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶))) ∧ ( oppFunc ‘(𝐶Δfunc𝐷)) ∈ (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶)))) → ( oppFunc ‘(𝐶Δfunc𝐷)) = ⟨(1st ‘( oppFunc ‘(𝐶Δfunc𝐷))), (2nd ‘( oppFunc ‘(𝐶Δfunc𝐷)))⟩)
57	53, 55, 56	sylancr 598	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → ( oppFunc ‘(𝐶Δfunc𝐷)) = ⟨(1st ‘( oppFunc ‘(𝐶Δfunc𝐷))), (2nd ‘( oppFunc ‘(𝐶Δfunc𝐷)))⟩)
58	57	oveq2d 7426	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → (⟨( oppFunc ↾ (𝐷 Func 𝐶)), (𝑓 ∈ (𝐷 Func 𝐶), 𝑔 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑔(𝐷 Nat 𝐶)𝑓)))⟩ ∘func ( oppFunc ‘(𝐶Δfunc𝐷))) = (⟨( oppFunc ↾ (𝐷 Func 𝐶)), (𝑓 ∈ (𝐷 Func 𝐶), 𝑔 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑔(𝐷 Nat 𝐶)𝑓)))⟩ ∘func ⟨(1st ‘( oppFunc ‘(𝐶Δfunc𝐷))), (2nd ‘( oppFunc ‘(𝐶Δfunc𝐷)))⟩))
59		relfunc 17914	. . . . . . . . . . 11 ⊢ Rel (𝑂 Func (𝑃 FuncCat 𝑂))
60		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑂Δfunc𝑃) = (𝑂Δfunc𝑃)
61	1	oppccat 17773	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
62	50, 61	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝑂 ∈ Cat)
63	8	oppccat 17773	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
64	49, 63	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝑃 ∈ Cat)
65	60, 62, 64, 15	diagcl 18292	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → (𝑂Δfunc𝑃) ∈ (𝑂 Func (𝑃 FuncCat 𝑂)))
66		1st2nd 8032	. . . . . . . . . . 11 ⊢ ((Rel (𝑂 Func (𝑃 FuncCat 𝑂)) ∧ (𝑂Δfunc𝑃) ∈ (𝑂 Func (𝑃 FuncCat 𝑂))) → (𝑂Δfunc𝑃) = ⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩)
67	59, 65, 66	sylancr 598	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → (𝑂Δfunc𝑃) = ⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩)
68	52, 58, 67	3eqtr3d 2806	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → (⟨( oppFunc ↾ (𝐷 Func 𝐶)), (𝑓 ∈ (𝐷 Func 𝐶), 𝑔 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑔(𝐷 Nat 𝐶)𝑓)))⟩ ∘func ⟨(1st ‘( oppFunc ‘(𝐶Δfunc𝐷))), (2nd ‘( oppFunc ‘(𝐶Δfunc𝐷)))⟩) = ⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩)
69	30	fucbas 18015	. . . . . . . . . 10 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
70	46, 69	oppcbas 17769	. . . . . . . . 9 ⊢ (𝐷 Func 𝐶) = (Base‘(oppCat‘(𝐷 FuncCat 𝐶)))
71	55	func1st2nd 49854	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → (1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))(𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶)))(2nd ‘( oppFunc ‘(𝐶Δfunc𝐷))))
72	43, 33	syl 18	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → ((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥) = ((1st ‘(𝐶Δfunc𝐷))‘𝑥))
73		simp2 1155	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝑥 ∈ (Base‘𝐶))
74		eqid 2763	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐶Δfunc𝐷))‘𝑥) = ((1st ‘(𝐶Δfunc𝐷))‘𝑥)
75	26, 50, 49, 20, 73, 74	diag1cl 18293	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → ((1st ‘(𝐶Δfunc𝐷))‘𝑥) ∈ (𝐷 Func 𝐶))
76	72, 75	eqeltrd 2863	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → ((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥) ∈ (𝐷 Func 𝐶))
77		eqidd 2764	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝑚 = 𝑚)
78		simp3 1156	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹))
79	48, 43, 76, 77, 78	opf2 50184	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → ((𝐹(𝑓 ∈ (𝐷 Func 𝐶), 𝑔 ∈ (𝐷 Func 𝐶) ↦ ( I ↾ (𝑔(𝐷 Nat 𝐶)𝑓)))((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥))‘𝑚) = 𝑚)
80		eqid 2763	. . . . . . . . 9 ⊢ (Hom ‘(oppCat‘(𝐷 FuncCat 𝐶))) = (Hom ‘(oppCat‘(𝐷 FuncCat 𝐶)))
81	30, 25	fuchom 18016	. . . . . . . . . . 11 ⊢ (𝐷 Nat 𝐶) = (Hom ‘(𝐷 FuncCat 𝐶))
82	81, 46	oppchom 17766	. . . . . . . . . 10 ⊢ (𝐹(Hom ‘(oppCat‘(𝐷 FuncCat 𝐶)))((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)) = (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)
83	78, 82	eleqtrrdi 2874	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝑚 ∈ (𝐹(Hom ‘(oppCat‘(𝐷 FuncCat 𝐶)))((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)))
84	45, 51, 68, 70, 43, 71, 79, 80, 83	uptr 49991	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → (𝑥(⟨(1st ‘( oppFunc ‘(𝐶Δfunc𝐷))), (2nd ‘( oppFunc ‘(𝐶Δfunc𝐷)))⟩(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚 ↔ 𝑥(⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚))
85	55	up1st2ndb 49965	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → (𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚 ↔ 𝑥(⟨(1st ‘( oppFunc ‘(𝐶Δfunc𝐷))), (2nd ‘( oppFunc ‘(𝐶Δfunc𝐷)))⟩(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚))
86	65	up1st2ndb 49965	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → (𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚 ↔ 𝑥(⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚))
87	84, 85, 86	3bitr4d 314	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → (𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚 ↔ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚))
88	19, 21, 42, 87	syl3anc 1398	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → (𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚 ↔ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚))
89	7, 88	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚)
90	6	up1st2nd 49963	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚) → 𝑥(⟨(1st ‘( oppFunc ‘(𝐶Δfunc𝐷))), (2nd ‘( oppFunc ‘(𝐶Δfunc𝐷)))⟩(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚)
91	90, 46, 69	oppcuprcl3 49978	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚) → 𝐹 ∈ (𝐷 Func 𝐶))
92	90, 1, 20	oppcuprcl4 49977	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚) → 𝑥 ∈ (Base‘𝐶))
93	90, 46, 81	oppcuprcl5 49979	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚) → 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹))
94	91, 92, 93, 87	syl3anc 1398	. . . . 5 ⊢ ((𝜑 ∧ 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚) → (𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚 ↔ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚))
95	6, 89, 94	bibiad 852	. . . 4 ⊢ (𝜑 → (𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚 ↔ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚))
96	4, 5, 95	eqbrrdiv 5780	. . 3 ⊢ (𝜑 → (( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺))
97	3, 96	eqtr3id 2812	. 2 ⊢ (𝜑 → (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺))
98		lmdfval2 50433	. 2 ⊢ ((𝐶 Limit 𝐷)‘𝐹) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
99		cmdfval2 50434	. 2 ⊢ ((𝑂 Colimit 𝑃)‘𝐺) = ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)
100	97, 98, 99	3eqtr4g 2823	1 ⊢ (𝜑 → ((𝐶 Limit 𝐷)‘𝐹) = ((𝑂 Colimit 𝑃)‘𝐺))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ⟨cop 4595   class class class wbr 5109   I cid 5555   ↾ cres 5663  Rel wrel 5666  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7980  2^nd c2nd 7981  Basecbs 17264  Hom chom 17316  Catccat 17715  oppCatcoppc 17762   Func cfunc 17906   ∘_func ccofu 17908   Nat cnat 17996   FuncCat cfuc 17997  Δ_funccdiag 18263   oppFunc coppf 49900   UP cup 49951   Limit clmd 50421   Colimit ccmd 50422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12858  df-fz 13531  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-hom 17329  df-cco 17330  df-cat 17719  df-cid 17720  df-homf 17721  df-comf 17722  df-oppc 17763  df-sect 17799  df-inv 17800  df-iso 17801  df-func 17910  df-idfu 17911  df-cofu 17912  df-full 17958  df-fth 17959  df-nat 17998  df-fuc 17999  df-catc 18151  df-xpc 18223  df-1stf 18224  df-curf 18265  df-diag 18267  df-oppf 49901  df-up 49952  df-lmd 50423  df-cmd 50424

This theorem is referenced by:  cmddu  50446