Theorem iscmd 49484

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 49476	. . . 4 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
2		islmd.l	. . . . 5 ⊢ 𝐿 = (𝐶Δfunc𝐷)
3	2	oveq1i 7413	. . . 4 ⊢ (𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
4	1, 3	eqtr4i 2761	. . 3 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = (𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
5	4	breqi 5125	. 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 49067	. . . . 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 49073	. . . 4 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑋 ∈ 𝐴)
10		eqid 2735	. . . . . 6 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
11		islmd.n	. . . . . 6 ⊢ 𝑁 = (𝐷 Nat 𝐶)
12	10, 11	fuchom 17975	. . . . 5 ⊢ 𝑁 = (Hom ‘(𝐷 FuncCat 𝐶))
13	7, 12	uprcl5 49074	. . . 4 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋)))
14	9, 13	jca 511	. . 3 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → (𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))))
15	11	natrcl 17964	. . . . . . . . . 10 ⊢ (𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋)) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶)))
16	15	adantl 481	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶)))
17	16	simpld 494	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐹 ∈ (𝐷 Func 𝐶))
18	17	func1st2nd 48991	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
19	18	funcrcl3 48993	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐶 ∈ Cat)
20	18	funcrcl2 48992	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐷 ∈ Cat)
21	2, 19, 20, 10	diagcl 18251	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
22	21	up1st2ndb 49069	. . . 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 17974	. . . . 5 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
24		islmd.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
25		eqid 2735	. . . . 5 ⊢ (comp‘(𝐷 FuncCat 𝐶)) = (comp‘(𝐷 FuncCat 𝐶))
26	21	func1st2nd 48991	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (1st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2nd ‘𝐿))
27		simpl 482	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝑋 ∈ 𝐴)
28		simpr 484	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋)))
29	8, 23, 24, 12, 25, 17, 26, 27, 28	isup 49063	. . . 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 726	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝐶 ∈ Cat)
32	20	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝐷 ∈ Cat)
33	27	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑋 ∈ 𝐴)
34		simplrl 776	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑥 ∈ 𝐴)
35		simpr 484	. . . . . . . . . 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 18255	. . . . . . . . 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 7418	. . . . . . . 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 726	. . . . . . . . 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 18256	. . . . . . . . 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 17976	. . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ Cat)
43	32	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ Cat)
44	33	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝑋 ∈ 𝐴)
45		eqid 2735	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
46		simpr 484	. . . . . . . . . . . . 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 18253	. . . . . . . . . . . 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 4856	. . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝑥 ∈ 𝐴)
50		eqid 2735	. . . . . . . . . . . 12 ⊢ ((1st ‘𝐿)‘𝑥) = ((1st ‘𝐿)‘𝑥)
51	2, 42, 43, 8, 49, 50, 30, 46	diag11 18253	. . . . . . . . . . 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 7421	. . . . . . . . . 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 3463	. . . . . . . . . . . 12 ⊢ 𝑚 ∈ V
54	53	fvconst2 7195	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝐵 → ((𝐵 × {𝑚})‘𝑗) = 𝑚)
55	54	adantl 481	. . . . . . . . . 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 2736	. . . . . . . . . 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 7424	. . . . . . . . 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 5214	. . . . . . . 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 2774	. . . . . . 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 2746	. . . . . 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 3375	. . . . 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 3203	. . . 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 305	. . 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 821	. 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 275	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃!wreu 3357  {csn 4601  ⟨cop 4607   class class class wbr 5119   ↦ cmpt 5201   × cxp 5652  ‘cfv 6530  (class class class)co 7403  1^st c1st 7984  2^nd c2nd 7985  Basecbs 17226  Hom chom 17280  compcco 17281  Catccat 17674   Func cfunc 17865   Nat cnat 17955   FuncCat cfuc 17956  Δ_funccdiag 18222   UP cup 49056   Colimit ccmd 49466

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8717  df-map 8840  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-nn 12239  df-2 12301  df-3 12302  df-4 12303  df-5 12304  df-6 12305  df-7 12306  df-8 12307  df-9 12308  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12851  df-fz 13523  df-struct 17164  df-slot 17199  df-ndx 17211  df-base 17227  df-hom 17293  df-cco 17294  df-cat 17678  df-cid 17679  df-func 17869  df-nat 17957  df-fuc 17958  df-xpc 18182  df-1stf 18183  df-curf 18224  df-diag 18226  df-up 49057  df-cmd 49468

This theorem is referenced by: (None)