Theorem iscmd 50657

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 50647	. . . 4 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
2		islmd.l	. . . . 5 ⊢ 𝐿 = (𝐶Δfunc𝐷)
3	2	oveq1i 7426	. . . 4 ⊢ (𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
4	1, 3	eqtr4i 2786	. . 3 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = (𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
5	4	breqi 5109	. 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 50176	. . . . 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 50182	. . . 4 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑋 ∈ 𝐴)
10		eqid 2760	. . . . . 6 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
11		islmd.n	. . . . . 6 ⊢ 𝑁 = (𝐷 Nat 𝐶)
12	10, 11	fuchom 18078	. . . . 5 ⊢ 𝑁 = (Hom ‘(𝐷 FuncCat 𝐶))
13	7, 12	uprcl5 50183	. . . 4 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋)))
14	9, 13	jca 521	. . 3 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → (𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))))
15	11	natrcl 18067	. . . . . . . . . 10 ⊢ (𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋)) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶)))
16	15	adantl 487	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ ((1st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶)))
17	16	simpld 500	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐹 ∈ (𝐷 Func 𝐶))
18	17	func1st2nd 50067	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
19	18	funcrcl3 50071	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐶 ∈ Cat)
20	18	funcrcl2 50070	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐷 ∈ Cat)
21	2, 19, 20, 10	diagcl 18354	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
22	21	up1st2ndb 50178	. . . 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 18077	. . . . 5 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
24		islmd.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
25		eqid 2760	. . . . 5 ⊢ (comp‘(𝐷 FuncCat 𝐶)) = (comp‘(𝐷 FuncCat 𝐶))
26	21	func1st2nd 50067	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → (1st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2nd ‘𝐿))
27		simpl 488	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝑋 ∈ 𝐴)
28		simpr 490	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) → 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋)))
29	8, 23, 24, 12, 25, 17, 26, 27, 28	isup 50171	. . . 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 739	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝐶 ∈ Cat)
32	20	ad2antrr 739	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝐷 ∈ Cat)
33	27	ad2antrr 739	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑋 ∈ 𝐴)
34		simplrl 789	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑥 ∈ 𝐴)
35		simpr 490	. . . . . . . . . 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 18358	. . . . . . . . 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 7431	. . . . . . . 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 739	. . . . . . . . 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 18359	. . . . . . . . 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 18079	. . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ Cat)
43	32	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ Cat)
44	33	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝑋 ∈ 𝐴)
45		eqid 2760	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
46		simpr 490	. . . . . . . . . . . . 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 18356	. . . . . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝑥 ∈ 𝐴)
50		eqid 2760	. . . . . . . . . . . 12 ⊢ ((1st ‘𝐿)‘𝑥) = ((1st ‘𝐿)‘𝑥)
51	2, 42, 43, 8, 49, 50, 30, 46	diag11 18356	. . . . . . . . . . 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 7434	. . . . . . . . . 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 3454	. . . . . . . . . . . 12 ⊢ 𝑚 ∈ V
54	53	fvconst2 7206	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝐵 → ((𝐵 × {𝑚})‘𝑗) = 𝑚)
55	54	adantl 487	. . . . . . . . . 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 2761	. . . . . . . . . 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 7437	. . . . . . . . 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 5198	. . . . . . . 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 2799	. . . . . . 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 2771	. . . . . 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 3379	. . . . 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 3224	. . . 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 834	. 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)))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃!wreu 3363  {csn 4584  ⟨cop 4590   class class class wbr 5103   ↦ cmpt 5186   × cxp 5653  ‘cfv 6535  (class class class)co 7416  1^st c1st 7990  2^nd c2nd 7991  Basecbs 17326  Hom chom 17378  compcco 17379  Catccat 17777   Func cfunc 17968   Nat cnat 18058   FuncCat cfuc 18059  Δ_funccdiag 18325   UP cup 50164   Colimit ccmd 50635

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  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 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-er 8703  df-map 8835  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11492  df-neg 11493  df-nn 12283  df-2 12352  df-3 12353  df-4 12354  df-5 12355  df-6 12356  df-7 12357  df-8 12358  df-9 12359  df-n0 12554  df-z 12641  df-dec 12762  df-uz 12913  df-fz 13587  df-struct 17264  df-slot 17299  df-ndx 17311  df-base 17327  df-hom 17391  df-cco 17392  df-cat 17781  df-cid 17782  df-func 17972  df-nat 18060  df-fuc 18061  df-xpc 18285  df-1stf 18286  df-curf 18327  df-diag 18329  df-up 50165  df-cmd 50637

This theorem is used by: (None)