Theorem lanup 49828

Description: The universal property of the left Kan extension; expressed explicitly. (Contributed by Zhi Wang, 4-Nov-2025.)

Hypotheses

Ref	Expression
lanup.s	⊢ 𝑆 = (𝐶 FuncCat 𝐸)
lanup.m	⊢ 𝑀 = (𝐷 Nat 𝐸)
lanup.n	⊢ 𝑁 = (𝐶 Nat 𝐸)
lanup.x	⊢ ∙ = (comp‘𝑆)
lanup.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
lanup.l	⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
lanup.a	⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)))

Assertion

Ref	Expression
lanup	⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴 ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴)))

Distinct variable groups:   ∙ ,𝑎,𝑏,𝑙   𝐴,𝑎,𝑏,𝑙   𝐶,𝑎,𝑏,𝑙   𝐷,𝑎,𝑏,𝑙   𝐸,𝑎,𝑏,𝑙   𝐹,𝑎,𝑏,𝑙   𝐿,𝑎,𝑏,𝑙   𝑀,𝑎,𝑏,𝑙   𝑁,𝑎,𝑏,𝑙   𝑆,𝑎,𝑏,𝑙   𝑋,𝑎,𝑏,𝑙   𝜑,𝑎,𝑏,𝑙

Proof of Theorem lanup

Step	Hyp	Ref	Expression
1		eqid 2734	. . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2	1	fucbas 17885	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
3		lanup.s	. . . 4 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
4	3	fucbas 17885	. . 3 ⊢ (𝐶 Func 𝐸) = (Base‘𝑆)
5		lanup.m	. . . 4 ⊢ 𝑀 = (𝐷 Nat 𝐸)
6	1, 5	fuchom 17886	. . 3 ⊢ 𝑀 = (Hom ‘(𝐷 FuncCat 𝐸))
7		lanup.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐸)
8	3, 7	fuchom 17886	. . 3 ⊢ 𝑁 = (Hom ‘𝑆)
9		lanup.x	. . 3 ⊢ ∙ = (comp‘𝑆)
10		lanup.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)))
11	7	natrcl 17875	. . . . 5 ⊢ (𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)) → (𝑋 ∈ (𝐶 Func 𝐸) ∧ (𝐿 ∘func 𝐹) ∈ (𝐶 Func 𝐸)))
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝐶 Func 𝐸) ∧ (𝐿 ∘func 𝐹) ∈ (𝐶 Func 𝐸)))
13	12	simpld 494	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
14	13	func1st2nd 49263	. . . . . 6 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
15	14	funcrcl3 49267	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
16		lanup.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
17	1, 15, 3, 16	prcoffunca 49573	. . . 4 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐸) Func 𝑆))
18	17	func1st2nd 49263	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))((𝐷 FuncCat 𝐸) Func 𝑆)(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
19		lanup.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
20		eqidd 2735	. . . . . 6 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
21	19, 20	prcof1 49575	. . . . 5 ⊢ (𝜑 → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿) = (𝐿 ∘func 𝐹))
22	21	oveq2d 7372	. . . 4 ⊢ (𝜑 → (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)) = (𝑋𝑁(𝐿 ∘func 𝐹)))
23	10, 22	eleqtrrd 2837	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)))
24	2, 4, 6, 8, 9, 13, 18, 19, 23	isup 49367	. 2 ⊢ (𝜑 → (𝐿(⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩((𝐷 FuncCat 𝐸) UP 𝑆)𝑋)𝐴 ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴)))
25		eqidd 2735	. . . . 5 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = (⟨𝐷, 𝐸⟩ −∘F 𝐹))
26	1, 3, 16, 13, 25	lanval 49806	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = ((⟨𝐷, 𝐸⟩ −∘F 𝐹)((𝐷 FuncCat 𝐸) UP 𝑆)𝑋))
27	26	breqd 5107	. . 3 ⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴 ↔ 𝐿((⟨𝐷, 𝐸⟩ −∘F 𝐹)((𝐷 FuncCat 𝐸) UP 𝑆)𝑋)𝐴))
28	17	up1st2ndb 49374	. . 3 ⊢ (𝜑 → (𝐿((⟨𝐷, 𝐸⟩ −∘F 𝐹)((𝐷 FuncCat 𝐸) UP 𝑆)𝑋)𝐴 ↔ 𝐿(⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩((𝐷 FuncCat 𝐸) UP 𝑆)𝑋)𝐴))
29	27, 28	bitrd 279	. 2 ⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴 ↔ 𝐿(⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩((𝐷 FuncCat 𝐸) UP 𝑆)𝑋)𝐴))
30		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → 𝑙 ∈ (𝐷 Func 𝐸))
31		eqidd 2735	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
32	30, 31	prcof1 49575	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙) = (𝑙 ∘func 𝐹))
33	32	eqcomd 2740	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (𝑙 ∘func 𝐹) = ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))
34	33	oveq2d 7372	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (𝑋𝑁(𝑙 ∘func 𝐹)) = (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙)))
35	21	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿) = (𝐿 ∘func 𝐹))
36	35	opeq2d 4834	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → ⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ = ⟨𝑋, (𝐿 ∘func 𝐹)⟩)
37	32	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙) = (𝑙 ∘func 𝐹))
38	36, 37	oveq12d 7374	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → (⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙)) = (⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹)))
39		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → 𝑏 ∈ (𝐿𝑀𝑙))
40		eqidd 2735	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
41	16	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → 𝐹 ∈ (𝐶 Func 𝐷))
42	5, 39, 40, 41	prcof21a 49578	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → ((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏) = (𝑏 ∘ (1st ‘𝐹)))
43		eqidd 2735	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → 𝐴 = 𝐴)
44	38, 42, 43	oveq123d 7377	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴) = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴))
45	44	eqcomd 2740	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴) = (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴))
46	45	eqeq2d 2745	. . . . 5 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → (𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴) ↔ 𝑎 = (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴)))
47	46	reubidva 3362	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) → (∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴) ↔ ∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴)))
48	34, 47	raleqbidva 3300	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (∀𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴) ↔ ∀𝑎 ∈ (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴)))
49	48	ralbidva 3155	. 2 ⊢ (𝜑 → (∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴) ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴)))
50	24, 29, 49	3bitr4d 311	1 ⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴 ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃!wreu 3346  ⟨cop 4584   class class class wbr 5096   ∘ ccom 5626  ‘cfv 6490  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  compcco 17187   Func cfunc 17776   ∘_func ccofu 17778   Nat cnat 17866   FuncCat cfuc 17867   UP cup 49360   −∘_F cprcof 49560   Lan clan 49792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8763  df-ixp 8834  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-z 12487  df-dec 12606  df-uz 12750  df-fz 13422  df-struct 17072  df-slot 17107  df-ndx 17119  df-base 17135  df-hom 17199  df-cco 17200  df-cat 17589  df-cid 17590  df-func 17780  df-cofu 17782  df-nat 17868  df-fuc 17869  df-xpc 18093  df-curf 18135  df-up 49361  df-swapf 49447  df-fuco 49504  df-prcof 49561  df-lan 49794

This theorem is referenced by: (None)