Theorem iscmd 50292

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 50282	. . . 4 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
2		islmd.l	. . . . 5 ⊢ 𝐿 = (𝐶Δfunc𝐷)
3	2	oveq1i 7408	. . . 4 ⊢ (𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
4	1, 3	eqtr4i 2790	. . 3 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = (𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
5	4	breqi 5108	. 2 ⊢ (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑅 ↔ 𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅)
6		id 22	. . . . . 6 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅)
7	6	up1st2nd 49811	. . . . 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 49817	. . . 4 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑋 ∈ 𝐴)
10		eqid 2764	. . . . . 6 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
11		islmd.n	. . . . . 6 ⊢ 𝑁 = (𝐷 Nat 𝐶)
12	10, 11	fuchom 17999	. . . . 5 ⊢ 𝑁 = (Hom ‘(𝐷 FuncCat 𝐶))
13	7, 12	uprcl5 49818	. . . 4 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋)))
14	9, 13	jca 519	. . 3 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → (𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))))
15	11	natrcl 17988	. . . . . . . . . 10 ⊢ (𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋)) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶)))
16	15	adantl 485	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶)))
17	16	simpld 498	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐹 ∈ (𝐷 Func 𝐶))
18	17	func1st2nd 49702	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
19	18	funcrcl3 49706	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐶 ∈ Cat)
20	18	funcrcl2 49705	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐷 ∈ Cat)
21	2, 19, 20, 10	diagcl 18275	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
22	21	up1st2ndb 49813	. . . 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 17998	. . . . 5 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
24		islmd.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
25		eqid 2764	. . . . 5 ⊢ (comp‘(𝐷 FuncCat 𝐶)) = (comp‘(𝐷 FuncCat 𝐶))
26	21	func1st2nd 49702	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (1st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2nd ‘𝐿))
27		simpl 486	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝑋 ∈ 𝐴)
28		simpr 488	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋)))
29	8, 23, 24, 12, 25, 17, 26, 27, 28	isup 49806	. . . 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 736	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝐶 ∈ Cat)
32	20	ad2antrr 736	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝐷 ∈ Cat)
33	27	ad2antrr 736	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑋 ∈ 𝐴)
34		simplrl 786	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑥 ∈ 𝐴)
35		simpr 488	. . . . . . . . . 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 18279	. . . . . . . . 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 7413	. . . . . . . 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 736	. . . . . . . . 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 18280	. . . . . . . . 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 18000	. . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ Cat)
43	32	adantr 484	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ Cat)
44	33	adantr 484	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝑋 ∈ 𝐴)
45		eqid 2764	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
46		simpr 488	. . . . . . . . . . . . 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 18277	. . . . . . . . . . . 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 4840	. . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝑥 ∈ 𝐴)
50		eqid 2764	. . . . . . . . . . . 12 ⊢ ((1st ‘𝐿)‘𝑥) = ((1st ‘𝐿)‘𝑥)
51	2, 42, 43, 8, 49, 50, 30, 46	diag11 18277	. . . . . . . . . . 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 7416	. . . . . . . . . 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 3460	. . . . . . . . . . . 12 ⊢ 𝑚 ∈ V
54	53	fvconst2 7190	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝐵 → ((𝐵 × {𝑚})‘𝑗) = 𝑚)
55	54	adantl 485	. . . . . . . . . 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 2765	. . . . . . . . . 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 7419	. . . . . . . . 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 5195	. . . . . . . 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 2803	. . . . . . 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 2775	. . . . . 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 3383	. . . . 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 3226	. . . 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 307	. . 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 831	. 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 277	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 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃!wreu 3367  {csn 4584  ⟨cop 4590   class class class wbr 5102   ↦ cmpt 5183   × cxp 5647  ‘cfv 6523  (class class class)co 7398  1^st c1st 7970  2^nd c2nd 7971  Basecbs 17247  Hom chom 17299  compcco 17300  Catccat 17698   Func cfunc 17889   Nat cnat 17979   FuncCat cfuc 17980  Δ_funccdiag 18246   UP cup 49799   Colimit ccmd 50270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-er 8680  df-map 8812  df-ixp 8882  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12484  df-z 12571  df-dec 12691  df-uz 12842  df-fz 13515  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17248  df-hom 17312  df-cco 17313  df-cat 17702  df-cid 17703  df-func 17893  df-nat 17981  df-fuc 17982  df-xpc 18206  df-1stf 18207  df-curf 18248  df-diag 18250  df-up 49800  df-cmd 50272

This theorem is referenced by: (None)