Theorem lmddu 49672

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 7363	. . . 4 ⊢ (𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶))) = ((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))
3	2	oveqi 7366	. . 3 ⊢ (( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
4		relup 49188	. . . 4 ⊢ Rel (( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
5		relup 49188	. . . 4 ⊢ Rel ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)
6		simpr 484	. . . . 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 484	. . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐷 ∈ 𝑊)
11		lmddu.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐶 ∈ 𝑉)
13		lmddu.g	. . . . . . . . 9 ⊢ 𝐺 = ( oppFunc ‘𝐹)
14	7	up1st2nd 49190	. . . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (𝑃 FuncCat 𝑂) = (𝑃 FuncCat 𝑂)
16	15	fucbas 17889	. . . . . . . . . 10 ⊢ (𝑃 Func 𝑂) = (Base‘(𝑃 FuncCat 𝑂))
17	14, 16	uprcl3 49195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐺 ∈ (𝑃 Func 𝑂))
18	13, 17	eqeltrrid 2833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → ( oppFunc ‘𝐹) ∈ (𝑃 Func 𝑂))
19	8, 1, 10, 12, 18	funcoppc5 49150	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐹 ∈ (𝐷 Func 𝐶))
20		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
21	14, 1, 20	oppcuprcl4 49204	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑥 ∈ (Base‘𝐶))
22		eqid 2729	. . . . . . . . . 10 ⊢ (𝑃 Nat 𝑂) = (𝑃 Nat 𝑂)
23	15, 22	fuchom 17890	. . . . . . . . 9 ⊢ (𝑃 Nat 𝑂) = (Hom ‘(𝑃 FuncCat 𝑂))
24	14, 23	uprcl5 49197	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑚 ∈ (𝐺(𝑃 Nat 𝑂)((1st ‘(𝑂Δfunc𝑃))‘𝑥)))
25		eqid 2729	. . . . . . . . 9 ⊢ (𝐷 Nat 𝐶) = (𝐷 Nat 𝐶)
26		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝐶Δfunc𝐷) = (𝐶Δfunc𝐷)
27		funcrcl 17789	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → (𝐷 ∈ Cat ∧ 𝐶 ∈ Cat))
28	27	simprd 495	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → 𝐶 ∈ Cat)
29	27	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → 𝐷 ∈ Cat)
30		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
31	26, 28, 29, 30	diagcl 18166	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → (𝐶Δfunc𝐷) ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
32	31	oppf1 49144	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → (1st ‘( oppFunc ‘(𝐶Δfunc𝐷))) = (1st ‘(𝐶Δfunc𝐷)))
33	32	fveq1d 6828	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → ((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥) = ((1st ‘(𝐶Δfunc𝐷))‘𝑥))
34	33	fveq2d 6830	. . . . . . . . . . 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 49420	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → ( oppFunc ‘((1st ‘(𝐶Δfunc𝐷))‘𝑥)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑥))
39	35, 38	eqtr2d 2765	. . . . . . . . 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 49236	. . . . . . . 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 2831	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹))
43		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝐹 ∈ (𝐷 Func 𝐶))
44	43	fvresd 6846	. . . . . . . . . 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 2782	. . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (oppCat‘(𝐷 FuncCat 𝐶)) = (oppCat‘(𝐷 FuncCat 𝐶))
47		eqidd 2730	. . . . . . . . . 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 2730	. . . . . . . . . 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 1133	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝐷 ∈ Cat)
50	28	3ad2ant1 1133	. . . . . . . . . 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 49416	. . . . . . . . 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 49421	. . . . . . . . . 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 17788	. . . . . . . . . . . 12 ⊢ Rel (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶)))
54	1, 46, 31	oppfoppc2 49147	. . . . . . . . . . . . 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 7981	. . . . . . . . . . . 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 587	. . . . . . . . . . 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 7369	. . . . . . . . . 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 17788	. . . . . . . . . . 11 ⊢ Rel (𝑂 Func (𝑃 FuncCat 𝑂))
60		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝑂Δfunc𝑃) = (𝑂Δfunc𝑃)
61	1	oppccat 17647	. . . . . . . . . . . . 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 17647	. . . . . . . . . . . . 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 18166	. . . . . . . . . . 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 7981	. . . . . . . . . . 11 ⊢ ((Rel (𝑂 Func (𝑃 FuncCat 𝑂)) ∧ (𝑂Δfunc𝑃) ∈ (𝑂 Func (𝑃 FuncCat 𝑂))) → (𝑂Δfunc𝑃) = ⟨(1st ‘(𝑂Δfunc𝑃)), (2nd ‘(𝑂Δfunc𝑃))⟩)
67	59, 65, 66	sylancr 587	. . . . . . . . . 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 2772	. . . . . . . . 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 17889	. . . . . . . . . 10 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
70	46, 69	oppcbas 17643	. . . . . . . . 9 ⊢ (𝐷 Func 𝐶) = (Base‘(oppCat‘(𝐷 FuncCat 𝐶)))
71	55	func1st2nd 49081	. . . . . . . . 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 1137	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝑥 ∈ (Base‘𝐶))
74		eqid 2729	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐶Δfunc𝐷))‘𝑥) = ((1st ‘(𝐶Δfunc𝐷))‘𝑥)
75	26, 50, 49, 20, 73, 74	diag1cl 18167	. . . . . . . . . . 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 2828	. . . . . . . . . 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 2730	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝑚 = 𝑚)
78		simp3 1138	. . . . . . . . . 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 49411	. . . . . . . . 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 2729	. . . . . . . . 9 ⊢ (Hom ‘(oppCat‘(𝐷 FuncCat 𝐶))) = (Hom ‘(oppCat‘(𝐷 FuncCat 𝐶)))
81	30, 25	fuchom 17890	. . . . . . . . . . 11 ⊢ (𝐷 Nat 𝐶) = (Hom ‘(𝐷 FuncCat 𝐶))
82	81, 46	oppchom 17640	. . . . . . . . . 10 ⊢ (𝐹(Hom ‘(oppCat‘(𝐷 FuncCat 𝐶)))((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)) = (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)
83	78, 82	eleqtrrdi 2839	. . . . . . . . 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 49218	. . . . . . . 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 49192	. . . . . . . 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 49192	. . . . . . . 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 311	. . . . . . 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 1373	. . . . . 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 257	. . . . 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 49190	. . . . . . 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 49205	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚) → 𝐹 ∈ (𝐷 Func 𝐶))
92	90, 1, 20	oppcuprcl4 49204	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚) → 𝑥 ∈ (Base‘𝐶))
93	90, 46, 81	oppcuprcl5 49206	. . . . . 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 1373	. . . . 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 839	. . . 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 5741	. . 3 ⊢ (𝜑 → (( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺))
97	3, 96	eqtr3id 2778	. 2 ⊢ (𝜑 → (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺))
98		lmdfval2 49660	. 2 ⊢ ((𝐶 Limit 𝐷)‘𝐹) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
99		cmdfval2 49661	. 2 ⊢ ((𝑂 Colimit 𝑃)‘𝐺) = ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)
100	97, 98, 99	3eqtr4g 2789	1 ⊢ (𝜑 → ((𝐶 Limit 𝐷)‘𝐹) = ((𝑂 Colimit 𝑃)‘𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ⟨cop 4585   class class class wbr 5095   I cid 5517   ↾ cres 5625  Rel wrel 5628  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  1^st c1st 7929  2^nd c2nd 7930  Basecbs 17139  Hom chom 17191  Catccat 17589  oppCatcoppc 17636   Func cfunc 17780   ∘_func ccofu 17782   Nat cnat 17870   FuncCat cfuc 17871  Δ_funccdiag 18137   oppFunc coppf 49127   UP cup 49178   Limit clmd 49648   Colimit ccmd 49649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  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 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-map 8762  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12611  df-uz 12755  df-fz 13430  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17140  df-hom 17204  df-cco 17205  df-cat 17593  df-cid 17594  df-homf 17595  df-comf 17596  df-oppc 17637  df-sect 17673  df-inv 17674  df-iso 17675  df-func 17784  df-idfu 17785  df-cofu 17786  df-full 17832  df-fth 17833  df-nat 17872  df-fuc 17873  df-catc 18025  df-xpc 18097  df-1stf 18098  df-curf 18139  df-diag 18141  df-oppf 49128  df-up 49179  df-lmd 49650  df-cmd 49651

This theorem is referenced by:  cmddu  49673