Theorem lmddu 50157

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 7366	. . . 4 ⊢ (𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶))) = ((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))
3	2	oveqi 7369	. . 3 ⊢ (( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
4		relup 49673	. . . 4 ⊢ Rel (( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
5		relup 49673	. . . 4 ⊢ Rel ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)
6		simpr 485	. . . . 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 485	. . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐷 ∈ 𝑊)
11		lmddu.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
12	11	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐶 ∈ 𝑉)
13		lmddu.g	. . . . . . . . 9 ⊢ 𝐺 = ( oppFunc ‘𝐹)
14	7	up1st2nd 49675	. . . . . . . . . 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 2739	. . . . . . . . . . 11 ⊢ (𝑃 FuncCat 𝑂) = (𝑃 FuncCat 𝑂)
16	15	fucbas 17921	. . . . . . . . . 10 ⊢ (𝑃 Func 𝑂) = (Base‘(𝑃 FuncCat 𝑂))
17	14, 16	uprcl3 49680	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐺 ∈ (𝑃 Func 𝑂))
18	13, 17	eqeltrrid 2844	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → ( oppFunc ‘𝐹) ∈ (𝑃 Func 𝑂))
19	8, 1, 10, 12, 18	funcoppc5 49635	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝐹 ∈ (𝐷 Func 𝐶))
20		eqid 2739	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
21	14, 1, 20	oppcuprcl4 49689	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑥 ∈ (Base‘𝐶))
22		eqid 2739	. . . . . . . . . 10 ⊢ (𝑃 Nat 𝑂) = (𝑃 Nat 𝑂)
23	15, 22	fuchom 17922	. . . . . . . . 9 ⊢ (𝑃 Nat 𝑂) = (Hom ‘(𝑃 FuncCat 𝑂))
24	14, 23	uprcl5 49682	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑚 ∈ (𝐺(𝑃 Nat 𝑂)((1st ‘(𝑂Δfunc𝑃))‘𝑥)))
25		eqid 2739	. . . . . . . . 9 ⊢ (𝐷 Nat 𝐶) = (𝐷 Nat 𝐶)
26		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (𝐶Δfunc𝐷) = (𝐶Δfunc𝐷)
27		funcrcl 17821	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → (𝐷 ∈ Cat ∧ 𝐶 ∈ Cat))
28	27	simprd 496	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → 𝐶 ∈ Cat)
29	27	simpld 495	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → 𝐷 ∈ Cat)
30		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
31	26, 28, 29, 30	diagcl 18198	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → (𝐶Δfunc𝐷) ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
32	31	oppf1 49629	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → (1st ‘( oppFunc ‘(𝐶Δfunc𝐷))) = (1st ‘(𝐶Δfunc𝐷)))
33	32	fveq1d 6829	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝐷 Func 𝐶) → ((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥) = ((1st ‘(𝐶Δfunc𝐷))‘𝑥))
34	33	fveq2d 6831	. . . . . . . . . . 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 49905	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → ( oppFunc ‘((1st ‘(𝐶Δfunc𝐷))‘𝑥)) = ((1st ‘(𝑂Δfunc𝑃))‘𝑥))
39	35, 38	eqtr2d 2775	. . . . . . . . 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 49721	. . . . . . . 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 2842	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)𝑚) → 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹))
43		simp1 1142	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝐹 ∈ (𝐷 Func 𝐶))
44	43	fvresd 6847	. . . . . . . . . 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 2792	. . . . . . . . 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 2739	. . . . . . . . . 10 ⊢ (oppCat‘(𝐷 FuncCat 𝐶)) = (oppCat‘(𝐷 FuncCat 𝐶))
47		eqidd 2740	. . . . . . . . . 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 2740	. . . . . . . . . 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 1139	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝐷 ∈ Cat)
50	28	3ad2ant1 1139	. . . . . . . . . 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 49901	. . . . . . . . 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 49906	. . . . . . . . . 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 17820	. . . . . . . . . . . 12 ⊢ Rel (𝑂 Func (oppCat‘(𝐷 FuncCat 𝐶)))
54	1, 46, 31	oppfoppc2 49632	. . . . . . . . . . . . 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 593	. . . . . . . . . . 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 7372	. . . . . . . . . 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 17820	. . . . . . . . . . 11 ⊢ Rel (𝑂 Func (𝑃 FuncCat 𝑂))
60		eqid 2739	. . . . . . . . . . . 12 ⊢ (𝑂Δfunc𝑃) = (𝑂Δfunc𝑃)
61	1	oppccat 17679	. . . . . . . . . . . . 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 17679	. . . . . . . . . . . . 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 18198	. . . . . . . . . . 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 593	. . . . . . . . . 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 2782	. . . . . . . . 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 17921	. . . . . . . . . 10 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
70	46, 69	oppcbas 17675	. . . . . . . . 9 ⊢ (𝐷 Func 𝐶) = (Base‘(oppCat‘(𝐷 FuncCat 𝐶)))
71	55	func1st2nd 49566	. . . . . . . . 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 1143	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝑥 ∈ (Base‘𝐶))
74		eqid 2739	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐶Δfunc𝐷))‘𝑥) = ((1st ‘(𝐶Δfunc𝐷))‘𝑥)
75	26, 50, 49, 20, 73, 74	diag1cl 18199	. . . . . . . . . . 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 2839	. . . . . . . . . 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 2740	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐷 Func 𝐶) ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑚 ∈ (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)) → 𝑚 = 𝑚)
78		simp3 1144	. . . . . . . . . 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 49896	. . . . . . . . 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 2739	. . . . . . . . 9 ⊢ (Hom ‘(oppCat‘(𝐷 FuncCat 𝐶))) = (Hom ‘(oppCat‘(𝐷 FuncCat 𝐶)))
81	30, 25	fuchom 17922	. . . . . . . . . . 11 ⊢ (𝐷 Nat 𝐶) = (Hom ‘(𝐷 FuncCat 𝐶))
82	81, 46	oppchom 17672	. . . . . . . . . 10 ⊢ (𝐹(Hom ‘(oppCat‘(𝐷 FuncCat 𝐶)))((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)) = (((1st ‘( oppFunc ‘(𝐶Δfunc𝐷)))‘𝑥)(𝐷 Nat 𝐶)𝐹)
83	78, 82	eleqtrrdi 2850	. . . . . . . . 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 49703	. . . . . . . 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 49677	. . . . . . . 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 49677	. . . . . . . 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 312	. . . . . . 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 1379	. . . . . 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 258	. . . . 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 49675	. . . . . . 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 49690	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚) → 𝐹 ∈ (𝐷 Func 𝐶))
92	90, 1, 20	oppcuprcl4 49689	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥(( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)𝑚) → 𝑥 ∈ (Base‘𝐶))
93	90, 46, 81	oppcuprcl5 49691	. . . . . 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 1379	. . . . 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 845	. . . 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 5737	. . 3 ⊢ (𝜑 → (( oppFunc ‘(𝐶Δfunc𝐷))(𝑂 UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺))
97	3, 96	eqtr3id 2788	. 2 ⊢ (𝜑 → (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹) = ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺))
98		lmdfval2 50145	. 2 ⊢ ((𝐶 Limit 𝐷)‘𝐹) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝐹)
99		cmdfval2 50146	. 2 ⊢ ((𝑂 Colimit 𝑃)‘𝐺) = ((𝑂Δfunc𝑃)(𝑂 UP (𝑃 FuncCat 𝑂))𝐺)
100	97, 98, 99	3eqtr4g 2799	1 ⊢ (𝜑 → ((𝐶 Limit 𝐷)‘𝐹) = ((𝑂 Colimit 𝑃)‘𝐺))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ⟨cop 4561   class class class wbr 5072   I cid 5512   ↾ cres 5620  Rel wrel 5623  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930  Basecbs 17170  Hom chom 17222  Catccat 17621  oppCatcoppc 17668   Func cfunc 17812   ∘_func ccofu 17814   Nat cnat 17902   FuncCat cfuc 17903  Δ_funccdiag 18169   oppFunc coppf 49612   UP cup 49663   Limit clmd 50133   Colimit ccmd 50134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  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 8633  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-cat 17625  df-cid 17626  df-homf 17627  df-comf 17628  df-oppc 17669  df-sect 17705  df-inv 17706  df-iso 17707  df-func 17816  df-idfu 17817  df-cofu 17818  df-full 17864  df-fth 17865  df-nat 17904  df-fuc 17905  df-catc 18057  df-xpc 18129  df-1stf 18130  df-curf 18171  df-diag 18173  df-oppf 49613  df-up 49664  df-lmd 50135  df-cmd 50136

This theorem is referenced by:  cmddu  50158