Theorem lmddu 50249

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 7401	. . . 4 ⊢ (𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶))) = ((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))
3	2	oveqi 7404	. . 3 ⊢ (( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
4		relup 49765	. . . 4 ⊢ Rel (( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
5		relup 49765	. . . 4 ⊢ Rel ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)
6		simpr 488	. . . . 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 488	. . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐷 ∈ 𝑊)
11		lmddu.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
12	11	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐶 ∈ 𝑉)
13		lmddu.g	. . . . . . . . 9 ⊢ 𝐺 = ( oppFunc ‘𝐹)
14	7	up1st2nd 49767	. . . . . . . . . 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 2761	. . . . . . . . . . 11 ⊢ (𝑃 FuncCat 𝑂) = (𝑃 FuncCat 𝑂)
16	15	fucbas 17987	. . . . . . . . . 10 ⊢ (𝑃 Func 𝑂) = (Base‘(𝑃 FuncCat 𝑂))
17	14, 16	uprcl3 49772	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐺 ∈ (𝑃 Func 𝑂))
18	13, 17	eqeltrrid 2866	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → ( oppFunc ‘𝐹) ∈ (𝑃 Func 𝑂))
19	8, 1, 10, 12, 18	funcoppc5 49727	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐹 ∈ (𝐷 Func 𝐶))
20		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
21	14, 1, 20	oppcuprcl4 49781	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑥 ∈ (Base‘𝐶))
22		eqid 2761	. . . . . . . . . 10 ⊢ (𝑃 Nat 𝑂) = (𝑃 Nat 𝑂)
23	15, 22	fuchom 17988	. . . . . . . . 9 ⊢ (𝑃 Nat 𝑂) = (Hom ‘(𝑃 FuncCat 𝑂))
24	14, 23	uprcl5 49774	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑚 ∈ (𝐺(𝑃 Nat 𝑂)((1st ‘(𝑂Δfunc𝑃))‘𝑥)))
25		eqid 2761	. . . . . . . . 9 ⊢ (𝐷 Nat 𝐶) = (𝐷 Nat 𝐶)
26		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (𝐶Δfunc𝐷) = (𝐶Δfunc𝐷)
27		funcrcl 17887	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → (𝐷 ∈ Cat ∧ 𝐶 ∈ Cat))
28	27	simprd 499	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → 𝐶 ∈ Cat)
29	27	simpld 498	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → 𝐷 ∈ Cat)
30		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
31	26, 28, 29, 30	diagcl 18264	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → (𝐶Δfunc𝐷) ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
32	31	oppf1 49721	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → (1st ‘( oppFunc ‘(𝐶Δfunc𝐷))) = (1st ‘(𝐶Δfunc𝐷)))
33	32	fveq1d 6864	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → ((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥) = ((1st ‘(𝐶Δfunc𝐷))‘𝑥))
34	33	fveq2d 6866	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → ( oppFunc ‘((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)) = ( oppFunc ‘((1st ‘(𝐶Δfunc𝐷))‘𝑥)))
35	19, 34	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → ( oppFunc ‘((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)) = ( oppFunc ‘((1st ‘(𝐶Δfunc𝐷))‘𝑥)))
36	19, 28	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐶 ∈ Cat)
37	19, 29	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐷 ∈ Cat)
38	1, 8, 26, 36, 37, 20, 21	oppfdiag1a 49997	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → ( oppFunc ‘((1st ‘(𝐶Δfunc𝐷))‘𝑥)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑥))
39	35, 38	eqtr2d 2797	. . . . . . . . 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 49813	. . . . . . . 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 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹))
43		simp1 1148	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝐹 ∈ (𝐷 Func 𝐶))
44	43	fvresd 6882	. . . . . . . . . 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 2814	. . . . . . . . 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 2761	. . . . . . . . . 10 ⊢ (oppCat‘(𝐷 FuncCat 𝐶)) = (oppCat‘(𝐷 FuncCat 𝐶))
47		eqidd 2762	. . . . . . . . . 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 2762	. . . . . . . . . 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 1145	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝐷 ∈ Cat)
50	28	3ad2ant1 1145	. . . . . . . . . 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 49993	. . . . . . . . 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 49998	. . . . . . . . . 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 17886	. . . . . . . . . . . 12 ⊢ Rel (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶)))
54	1, 46, 31	oppfoppc2 49724	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → ( oppFunc ‘(𝐶Δfunc𝐷)) ∈ (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶))))
55	43, 54	syl 17	. . . . . . . . . . . 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 8015	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 7407	. . . . . . . . . 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 17886	. . . . . . . . . . 11 ⊢ Rel (𝑂 Func (𝑃 FuncCat 𝑂))
60		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑂Δfunc𝑃) = (𝑂Δfunc𝑃)
61	1	oppccat 17745	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
62	50, 61	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝑂 ∈ Cat)
63	8	oppccat 17745	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
64	49, 63	syl 17	. . . . . . . . . . . 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 18264	. . . . . . . . . . 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 8015	. . . . . . . . . . 11 ⊢ ((Rel (𝑂 Func (𝑃 FuncCat 𝑂)) ∧ (𝑂Δfunc𝑃) ∈ (𝑂 Func (𝑃 FuncCat 𝑂))) → (𝑂Δfunc𝑃) = ⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩)
67	59, 65, 66	sylancr 596	. . . . . . . . . 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 2804	. . . . . . . . 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 17987	. . . . . . . . . 10 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
70	46, 69	oppcbas 17741	. . . . . . . . 9 ⊢ (𝐷 Func 𝐶) = (Base‘(oppCat‘(𝐷 FuncCat 𝐶)))
71	55	func1st2nd 49658	. . . . . . . . 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 17	. . . . . . . . . . 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 1149	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝑥 ∈ (Base‘𝐶))
74		eqid 2761	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐶Δfunc𝐷))‘𝑥) = ((1st ‘(𝐶Δfunc𝐷))‘𝑥)
75	26, 50, 49, 20, 73, 74	diag1cl 18265	. . . . . . . . . . 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 2861	. . . . . . . . . 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 2762	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝑚 = 𝑚)
78		simp3 1150	. . . . . . . . . 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 49988	. . . . . . . . 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 2761	. . . . . . . . 9 ⊢ (Hom ‘(oppCat‘(𝐷 FuncCat 𝐶))) = (Hom ‘(oppCat‘(𝐷 FuncCat 𝐶)))
81	30, 25	fuchom 17988	. . . . . . . . . . 11 ⊢ (𝐷 Nat 𝐶) = (Hom ‘(𝐷 FuncCat 𝐶))
82	81, 46	oppchom 17738	. . . . . . . . . 10 ⊢ (𝐹(Hom ‘(oppCat‘(𝐷 FuncCat 𝐶)))((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)) = (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)
83	78, 82	eleqtrrdi 2872	. . . . . . . . 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 49795	. . . . . . . 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 49769	. . . . . . . 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 49769	. . . . . . . 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 313	. . . . . . 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 1389	. . . . . 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 259	. . . . 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 49767	. . . . . . 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 49782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚) → 𝐹 ∈ (𝐷 Func 𝐶))
92	90, 1, 20	oppcuprcl4 49781	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚) → 𝑥 ∈ (Base‘𝐶))
93	90, 46, 81	oppcuprcl5 49783	. . . . . 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 1389	. . . . 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 850	. . . 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 5762	. . 3 ⊢ (𝜑 → (( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺))
97	3, 96	eqtr3id 2810	. 2 ⊢ (𝜑 → (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺))
98		lmdfval2 50237	. 2 ⊢ ((𝐶 Limit 𝐷)‘𝐹) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
99		cmdfval2 50238	. 2 ⊢ ((𝑂 Colimit 𝑃)‘𝐺) = ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)
100	97, 98, 99	3eqtr4g 2821	1 ⊢ (𝜑 → ((𝐶 Limit 𝐷)‘𝐹) = ((𝑂 Colimit 𝑃)‘𝐺))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ⟨cop 4585   class class class wbr 5097   I cid 5537   ↾ cres 5645  Rel wrel 5648  ‘cfv 6516  (class class class)co 7391   ∈ cmpo 7393  1^st c1st 7963  2^nd c2nd 7964  Basecbs 17236  Hom chom 17288  Catccat 17687  oppCatcoppc 17734   Func cfunc 17878   ∘_func ccofu 17880   Nat cnat 17968   FuncCat cfuc 17969  Δ_funccdiag 18235   oppFunc coppf 49704   UP cup 49755   Limit clmd 50225   Colimit ccmd 50226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-tpos 8200  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-er 8672  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12476  df-z 12563  df-dec 12683  df-uz 12834  df-fz 13507  df-struct 17174  df-sets 17191  df-slot 17209  df-ndx 17221  df-base 17237  df-hom 17301  df-cco 17302  df-cat 17691  df-cid 17692  df-homf 17693  df-comf 17694  df-oppc 17735  df-sect 17771  df-inv 17772  df-iso 17773  df-func 17882  df-idfu 17883  df-cofu 17884  df-full 17930  df-fth 17931  df-nat 17970  df-fuc 17971  df-catc 18123  df-xpc 18195  df-1stf 18196  df-curf 18237  df-diag 18239  df-oppf 49705  df-up 49756  df-lmd 50227  df-cmd 50228

This theorem is referenced by:  cmddu  50250