iscmd - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > iscmd		Structured version Visualization version GIF version

Theorem iscmd 49659

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

**Hypotheses**
Ref	Expression
islmd.l	⊢ 𝐿 = (𝐶Δ_func𝐷)
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((1^st ‘𝐿)‘𝑋))) ∧ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1^st ‘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 49649	. . . 4 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = ((𝐶Δ_func𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
2		islmd.l	. . . . 5 ⊢ 𝐿 = (𝐶Δ_func𝐷)
3	2	oveq1i 7400	. . . 4 ⊢ (𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹) = ((𝐶Δ_func𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
4	1, 3	eqtr4i 2756	. . 3 ⊢ ((𝐶 Colimit 𝐷)‘𝐹) = (𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)
5	4	breqi 5116	. 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 49178	. . . . 5 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑋(⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅)
8		islmd.a	. . . . 5 ⊢ 𝐴 = (Base‘𝐶)
9	7, 8	uprcl4 49184	. . . 4 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑋 ∈ 𝐴)
10		eqid 2730	. . . . . 6 ⊢ (𝐷 FuncCat 𝐶) = (𝐷 FuncCat 𝐶)
11		islmd.n	. . . . . 6 ⊢ 𝑁 = (𝐷 Nat 𝐶)
12	10, 11	fuchom 17933	. . . . 5 ⊢ 𝑁 = (Hom ‘(𝐷 FuncCat 𝐶))
13	7, 12	uprcl5 49185	. . . 4 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋)))
14	9, 13	jca 511	. . 3 ⊢ (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 → (𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))))
15	11	natrcl 17922	. . . . . . . . . 10 ⊢ (𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋)) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ ((1^st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶)))
16	15	adantl 481	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) → (𝐹 ∈ (𝐷 Func 𝐶) ∧ ((1^st ‘𝐿)‘𝑋) ∈ (𝐷 Func 𝐶)))
17	16	simpld 494	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) → 𝐹 ∈ (𝐷 Func 𝐶))
18	17	func1st2nd 49069	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) → (1^st ‘𝐹)(𝐷 Func 𝐶)(2^nd ‘𝐹))
19	18	funcrcl3 49073	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) → 𝐶 ∈ Cat)
20	18	funcrcl2 49072	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) → 𝐷 ∈ Cat)
21	2, 19, 20, 10	diagcl 18209	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) → 𝐿 ∈ (𝐶 Func (𝐷 FuncCat 𝐶)))
22	21	up1st2ndb 49180	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) → (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 ↔ 𝑋(⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅))
23	10	fucbas 17932	. . . . 5 ⊢ (𝐷 Func 𝐶) = (Base‘(𝐷 FuncCat 𝐶))
24		islmd.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
25		eqid 2730	. . . . 5 ⊢ (comp‘(𝐷 FuncCat 𝐶)) = (comp‘(𝐷 FuncCat 𝐶))
26	21	func1st2nd 49069	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) → (1^st ‘𝐿)(𝐶 Func (𝐷 FuncCat 𝐶))(2^nd ‘𝐿))
27		simpl 482	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) → 𝑋 ∈ 𝐴)
28		simpr 484	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) → 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋)))
29	8, 23, 24, 12, 25, 17, 26, 27, 28	isup 49173	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) → (𝑋(⟨(1^st ‘𝐿), (2^nd ‘𝐿)⟩(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (((𝑋(2^nd ‘𝐿)𝑥)‘𝑚)(⟨𝐹, ((1^st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1^st ‘𝐿)‘𝑥))𝑅)))
30		islmd.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐷)
31	19	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝐶 ∈ Cat)
32	20	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝐷 ∈ Cat)
33	27	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑋 ∈ 𝐴)
34		simplrl 776	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑥 ∈ 𝐴)
35		simpr 484	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑚 ∈ (𝑋𝐻𝑥))
36	2, 8, 30, 24, 31, 32, 33, 34, 35	diag2 18213	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → ((𝑋(2^nd ‘𝐿)𝑥)‘𝑚) = (𝐵 × {𝑚}))
37	36	oveq1d 7405	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → (((𝑋(2^nd ‘𝐿)𝑥)‘𝑚)(⟨𝐹, ((1^st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1^st ‘𝐿)‘𝑥))𝑅) = ((𝐵 × {𝑚})(⟨𝐹, ((1^st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1^st ‘𝐿)‘𝑥))𝑅))
38		islmd.x	. . . . . . . . 9 ⊢ · = (comp‘𝐶)
39	28	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋)))
40	2, 8, 30, 24, 31, 32, 33, 34, 35, 11	diag2cl 18214	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → (𝐵 × {𝑚}) ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑥)))
41	10, 11, 30, 38, 25, 39, 40	fucco 17934	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → ((𝐵 × {𝑚})(⟨𝐹, ((1^st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1^st ‘𝐿)‘𝑥))𝑅) = (𝑗 ∈ 𝐵 ↦ (((𝐵 × {𝑚})‘𝑗)(⟨((1^st ‘𝐹)‘𝑗), ((1^st ‘((1^st ‘𝐿)‘𝑋))‘𝑗)⟩ · ((1^st ‘((1^st ‘𝐿)‘𝑥))‘𝑗))(𝑅‘𝑗))))
42	31	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ Cat)
43	32	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ Cat)
44	33	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝑋 ∈ 𝐴)
45		eqid 2730	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝐿)‘𝑋) = ((1^st ‘𝐿)‘𝑋)
46		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝑗 ∈ 𝐵)
47	2, 42, 43, 8, 44, 45, 30, 46	diag11 18211	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → ((1^st ‘((1^st ‘𝐿)‘𝑋))‘𝑗) = 𝑋)
48	47	opeq2d 4847	. . . . . . . . . . 11 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → ⟨((1^st ‘𝐹)‘𝑗), ((1^st ‘((1^st ‘𝐿)‘𝑋))‘𝑗)⟩ = ⟨((1^st ‘𝐹)‘𝑗), 𝑋⟩)
49	34	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → 𝑥 ∈ 𝐴)
50		eqid 2730	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐿)‘𝑥) = ((1^st ‘𝐿)‘𝑥)
51	2, 42, 43, 8, 49, 50, 30, 46	diag11 18211	. . . . . . . . . . 11 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → ((1^st ‘((1^st ‘𝐿)‘𝑥))‘𝑗) = 𝑥)
52	48, 51	oveq12d 7408	. . . . . . . . . 10 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → (⟨((1^st ‘𝐹)‘𝑗), ((1^st ‘((1^st ‘𝐿)‘𝑋))‘𝑗)⟩ · ((1^st ‘((1^st ‘𝐿)‘𝑥))‘𝑗)) = (⟨((1^st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥))
53		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑚 ∈ V
54	53	fvconst2 7181	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝐵 → ((𝐵 × {𝑚})‘𝑗) = 𝑚)
55	54	adantl 481	. . . . . . . . . 10 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → ((𝐵 × {𝑚})‘𝑗) = 𝑚)
56		eqidd 2731	. . . . . . . . . 10 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → (𝑅‘𝑗) = (𝑅‘𝑗))
57	52, 55, 56	oveq123d 7411	. . . . . . . . 9 ⊢ (((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) ∧ 𝑗 ∈ 𝐵) → (((𝐵 × {𝑚})‘𝑗)(⟨((1^st ‘𝐹)‘𝑗), ((1^st ‘((1^st ‘𝐿)‘𝑋))‘𝑗)⟩ · ((1^st ‘((1^st ‘𝐿)‘𝑥))‘𝑗))(𝑅‘𝑗)) = (𝑚(⟨((1^st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))
58	57	mpteq2dva 5203	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → (𝑗 ∈ 𝐵 ↦ (((𝐵 × {𝑚})‘𝑗)(⟨((1^st ‘𝐹)‘𝑗), ((1^st ‘((1^st ‘𝐿)‘𝑋))‘𝑗)⟩ · ((1^st ‘((1^st ‘𝐿)‘𝑥))‘𝑗))(𝑅‘𝑗))) = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1^st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗))))
59	37, 41, 58	3eqtrd 2769	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → (((𝑋(2^nd ‘𝐿)𝑥)‘𝑚)(⟨𝐹, ((1^st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1^st ‘𝐿)‘𝑥))𝑅) = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1^st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗))))
60	59	eqeq2d 2741	. . . . . 6 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) ∧ 𝑚 ∈ (𝑋𝐻𝑥)) → (𝑎 = (((𝑋(2^nd ‘𝐿)𝑥)‘𝑚)(⟨𝐹, ((1^st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1^st ‘𝐿)‘𝑥))𝑅) ↔ 𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1^st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))
61	60	reubidva 3372	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ (𝑥 ∈ 𝐴 ∧ 𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥)))) → (∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (((𝑋(2^nd ‘𝐿)𝑥)‘𝑚)(⟨𝐹, ((1^st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1^st ‘𝐿)‘𝑥))𝑅) ↔ ∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1^st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))
62	61	2ralbidva 3200	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) → (∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (((𝑋(2^nd ‘𝐿)𝑥)‘𝑚)(⟨𝐹, ((1^st ‘𝐿)‘𝑋)⟩(comp‘(𝐷 FuncCat 𝐶))((1^st ‘𝐿)‘𝑥))𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1^st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))
63	22, 29, 62	3bitrd 305	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) → (𝑋(𝐿(𝐶 UP (𝐷 FuncCat 𝐶))𝐹)𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1^st ‘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((1^st ‘𝐿)‘𝑋))) ∧ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1^st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))
65	5, 64	bitri 275	1 ⊢ (𝑋((𝐶 Colimit 𝐷)‘𝐹)𝑅 ↔ ((𝑋 ∈ 𝐴 ∧ 𝑅 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑋))) ∧ ∀𝑥 ∈ 𝐴 ∀𝑎 ∈ (𝐹𝑁((1^st ‘𝐿)‘𝑥))∃!𝑚 ∈ (𝑋𝐻𝑥)𝑎 = (𝑗 ∈ 𝐵 ↦ (𝑚(⟨((1^st ‘𝐹)‘𝑗), 𝑋⟩ · 𝑥)(𝑅‘𝑗)))))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∃!wreu 3354 {csn 4592 ⟨cop 4598 class class class wbr 5110 ↦ cmpt 5191 × cxp 5639 ‘cfv 6514 (class class class)co 7390 1^st c1st 7969 2^nd c2nd 7970 Basecbs 17186 Hom chom 17238 compcco 17239 Catccat 17632 Func cfunc 17823 Nat cnat 17913 FuncCat cfuc 17914 Δ_funccdiag 18180 UP cup 49166 Colimit ccmd 49637

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 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 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 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-fz 13476 df-struct 17124 df-slot 17159 df-ndx 17171 df-base 17187 df-hom 17251 df-cco 17252 df-cat 17636 df-cid 17637 df-func 17827 df-nat 17915 df-fuc 17916 df-xpc 18140 df-1stf 18141 df-curf 18182 df-diag 18184 df-up 49167 df-cmd 49639

This theorem is referenced by: (None)

W3C validator