Theorem iscmd 50366

Description: The universal property of colimits of a diagram. (Contributed by Zhi Wang, 13-Nov-2025.)

Hypotheses

Ref	Expression
islmd.l	⊢ 𝐿 = (𝐶Δfunc𝐷)
islmd.a	⊢ 𝐴 = (Base‘𝐶)
islmd.n	⊢ 𝑁 = (𝐷 Nat 𝐶)
islmd.b	⊢ 𝐵 = (Base‘𝐷)
islmd.h	⊢ 𝐻 = (Hom ‘𝐶)
islmd.x	⊢ · = (comp‘𝐶)

Assertion

Ref	Expression
iscmd	⊢ (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑅 ↔ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))

Distinct variable groups:   · ,𝑗   𝐴,𝑎,𝑗,𝑚,𝑥   𝐵,𝑗   𝐶,𝑎,𝑗,𝑚,𝑥   𝐷,𝑎,𝑗,𝑚,𝑥   𝐹,𝑎,𝑗,𝑚,𝑥   𝑗,𝐻,𝑚   𝐿,𝑎,𝑗,𝑚,𝑥   𝑁,𝑎,𝑗,𝑚,𝑥   𝑅,𝑎,𝑗,𝑚,𝑥   𝑋,𝑎,𝑗,𝑚,𝑥   𝐻,𝑎,𝑥

Allowed substitution hints:   𝐵(𝑥,𝑚,𝑎)   · (𝑥,𝑚,𝑎)

Proof of Theorem iscmd

Step	Hyp	Ref	Expression
1		cmdfval2 50356	. . . 4 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
2		islmd.l	. . . . 5 ⊢ 𝐿 = (𝐶Δfunc𝐷)
3	2	oveq1i 7423	. . . 4 ⊢ (𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
4	1, 3	eqtr4i 2795	. . 3 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = (𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
5	4	breqi 5119	. 2 ⊢ (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑅 ↔ 𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅)
6		id 23	. . . . . 6 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅)
7	6	up1st2nd 49885	. . . . 5 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑋(⟨(1st ‘𝐿), (2nd ‘𝐿)⟩(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅)
8		islmd.a	. . . . 5 ⊢ 𝐴 = (Base‘𝐶)
9	7, 8	uprcl4 49891	. . . 4 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑋 ∈ 𝐴)
10		eqid 2769	. . . . . 6 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
11		islmd.n	. . . . . 6 ⊢ 𝑁 = (𝐷 Nat 𝐶)
12	10, 11	fuchom 18023	. . . . 5 ⊢ 𝑁 = (Hom ‘(𝐷 FuncCat 𝐶))
13	7, 12	uprcl5 49892	. . . 4 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋)))
14	9, 13	jca 520	. . 3 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → (𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))))
15	11	natrcl 18012	. . . . . . . . . 10 ⊢ (𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋)) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶)))
16	15	adantl 486	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶)))
17	16	simpld 499	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐹 ∈ (𝐷 Func 𝐶))
18	17	func1st2nd 49776	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
19	18	funcrcl3 49780	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐶 ∈ Cat)
20	18	funcrcl2 49779	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐷 ∈ Cat)
21	2, 19, 20, 10	diagcl 18299	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
22	21	up1st2ndb 49887	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 ↔ 𝑋(⟨(1st ‘𝐿), (2nd ‘𝐿)⟩(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅))
23	10	fucbas 18022	. . . . 5 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
24		islmd.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
25		eqid 2769	. . . . 5 ⊢ (comp‘(𝐷 FuncCat 𝐶)) = (comp‘(𝐷 FuncCat 𝐶))
26	21	func1st2nd 49776	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (1st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2nd ‘𝐿))
27		simpl 487	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝑋 ∈ 𝐴)
28		simpr 489	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋)))
29	8, 23, 24, 12, 25, 17, 26, 27, 28	isup 49880	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (𝑋(⟨(1st ‘𝐿), (2nd ‘𝐿)⟩(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (((𝑋(2nd ‘𝐿)𝑥)‘𝑚)(⟨𝐹, ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1st ‘𝐿)‘𝑥))𝑅)))
30		islmd.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐷)
31	19	ad2antrr 738	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝐶 ∈ Cat)
32	20	ad2antrr 738	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝐷 ∈ Cat)
33	27	ad2antrr 738	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑋 ∈ 𝐴)
34		simplrl 788	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑥 ∈ 𝐴)
35		simpr 489	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑚 ∈ (𝑋𝐻𝑥))
36	2, 8, 30, 24, 31, 32, 33, 34, 35	diag2 18303	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → ((𝑋(2nd ‘𝐿)𝑥)‘𝑚) = (𝐵 × {𝑚}))
37	36	oveq1d 7428	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → (((𝑋(2nd ‘𝐿)𝑥)‘𝑚)(⟨𝐹, ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1st ‘𝐿)‘𝑥))𝑅) = ((𝐵 × {𝑚})(⟨𝐹, ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1st ‘𝐿)‘𝑥))𝑅))
38		islmd.x	. . . . . . . . 9 ⊢ · = (comp‘𝐶)
39	28	ad2antrr 738	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋)))
40	2, 8, 30, 24, 31, 32, 33, 34, 35, 11	diag2cl 18304	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → (𝐵 × {𝑚}) ∈ (((1st ‘𝐿)‘𝑋)𝑁((1st ‘𝐿)‘𝑥)))
41	10, 11, 30, 38, 25, 39, 40	fucco 18024	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → ((𝐵 × {𝑚})(⟨𝐹, ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1st ‘𝐿)‘𝑥))𝑅) = (𝑗 ∈ 𝐵 ↦ (((𝐵 × {𝑚})‘𝑗)(⟨((1st ‘𝐹)‘𝑗), ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑗)⟩ · ((1st ‘((1st ‘𝐿)‘𝑥))‘𝑗))(𝑅‘𝑗))))
42	31	adantr 485	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ Cat)
43	32	adantr 485	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ Cat)
44	33	adantr 485	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝑋 ∈ 𝐴)
45		eqid 2769	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
46		simpr 489	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝑗 ∈ 𝐵)
47	2, 42, 43, 8, 44, 45, 30, 46	diag11 18301	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑗) = 𝑋)
48	47	opeq2d 4849	. . . . . . . . . . 11 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → ⟨((1st ‘𝐹)‘𝑗), ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑗)⟩ = ⟨((1st ‘𝐹)‘𝑗), 𝑋⟩)
49	34	adantr 485	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝑥 ∈ 𝐴)
50		eqid 2769	. . . . . . . . . . . 12 ⊢ ((1st ‘𝐿)‘𝑥) = ((1st ‘𝐿)‘𝑥)
51	2, 42, 43, 8, 49, 50, 30, 46	diag11 18301	. . . . . . . . . . 11 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → ((1st ‘((1st ‘𝐿)‘𝑥))‘𝑗) = 𝑥)
52	48, 51	oveq12d 7431	. . . . . . . . . 10 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → (⟨((1st ‘𝐹)‘𝑗), ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑗)⟩ · ((1st ‘((1st ‘𝐿)‘𝑥))‘𝑗)) = (⟨((1st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥))
53		vex 3467	. . . . . . . . . . . 12 ⊢ 𝑚 ∈ V
54	53	fvconst2 7205	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝐵 → ((𝐵 × {𝑚})‘𝑗) = 𝑚)
55	54	adantl 486	. . . . . . . . . 10 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → ((𝐵 × {𝑚})‘𝑗) = 𝑚)
56		eqidd 2770	. . . . . . . . . 10 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → (𝑅‘𝑗) = (𝑅‘𝑗))
57	52, 55, 56	oveq123d 7434	. . . . . . . . 9 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → (((𝐵 × {𝑚})‘𝑗)(⟨((1st ‘𝐹)‘𝑗), ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑗)⟩ · ((1st ‘((1st ‘𝐿)‘𝑥))‘𝑗))(𝑅‘𝑗)) = (𝑚(⟨((1st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))
58	57	mpteq2dva 5208	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → (𝑗 ∈ 𝐵 ↦ (((𝐵 × {𝑚})‘𝑗)(⟨((1st ‘𝐹)‘𝑗), ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑗)⟩ · ((1st ‘((1st ‘𝐿)‘𝑥))‘𝑗))(𝑅‘𝑗))) = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗))))
59	37, 41, 58	3eqtrd 2808	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → (((𝑋(2nd ‘𝐿)𝑥)‘𝑚)(⟨𝐹, ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1st ‘𝐿)‘𝑥))𝑅) = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗))))
60	59	eqeq2d 2780	. . . . . 6 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → (𝑎 = (((𝑋(2nd ‘𝐿)𝑥)‘𝑚)(⟨𝐹, ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1st ‘𝐿)‘𝑥))𝑅) ↔ 𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))
61	60	reubidva 3390	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) → (∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (((𝑋(2nd ‘𝐿)𝑥)‘𝑚)(⟨𝐹, ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1st ‘𝐿)‘𝑥))𝑅) ↔ ∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))
62	61	2ralbidva 3233	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (((𝑋(2nd ‘𝐿)𝑥)‘𝑚)(⟨𝐹, ((1st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1st ‘𝐿)‘𝑥))𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))
63	22, 29, 62	3bitrd 308	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))
64	14, 63	biadanii 833	. 2 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 ↔ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))
65	5, 64	bitri 278	1 ⊢ (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑅 ↔ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃!wreu 3374  {csn 4594  ⟨cop 4600   class class class wbr 5113   ↦ cmpt 5196   × cxp 5662  ‘cfv 6539  (class class class)co 7413  1^st c1st 7986  2^nd c2nd 7987  Basecbs 17271  Hom chom 17323  compcco 17324  Catccat 17722   Func cfunc 17913   Nat cnat 18003   FuncCat cfuc 18004  Δ_funccdiag 18270   UP cup 49873   Colimit ccmd 50344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735  ax-cnex 11158  ax-resscn 11159  ax-1cn 11160  ax-icn 11161  ax-addcl 11162  ax-addrcl 11163  ax-mulcl 11164  ax-mulrcl 11165  ax-mulcom 11166  ax-addass 11167  ax-mulass 11168  ax-distr 11169  ax-i2m1 11170  ax-1ne0 11171  ax-1rid 11172  ax-rnegex 11173  ax-rrecex 11174  ax-cnre 11175  ax-pre-lttri 11176  ax-pre-lttrn 11177  ax-pre-ltadd 11178  ax-pre-mulgt0 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11445  df-neg 11446  df-nn 12236  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12865  df-fz 13538  df-struct 17209  df-slot 17244  df-ndx 17256  df-base 17272  df-hom 17336  df-cco 17337  df-cat 17726  df-cid 17727  df-func 17917  df-nat 18005  df-fuc 18006  df-xpc 18230  df-1stf 18231  df-curf 18272  df-diag 18274  df-up 49874  df-cmd 50346

This theorem is referenced by: (None)