Theorem ranup 50140

Description: The universal property of the right Kan extension; expressed explicitly. (Contributed by Zhi Wang, 5-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))
ranup.a	⊢ (𝜑 → 𝐴 ∈ ((𝐿 ∘func 𝐹)𝑁𝑋))

Assertion

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

Distinct variable groups:   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,𝑏   𝑁,𝑎,𝑏   𝑆,𝑎,𝑏,𝑙   𝑋,𝑎,𝑏,𝑙   𝜑,𝑎,𝑏,𝑙

Allowed substitution hints:   ∙ (𝑎,𝑏,𝑙)   𝑀(𝑎,𝑙)   𝑁(𝑙)

Proof of Theorem ranup

Step	Hyp	Ref	Expression
1		eqid 2739	. . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2	1	fucbas 17922	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
3		lanup.s	. . . 4 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
4	3	fucbas 17922	. . 3 ⊢ (𝐶 Func 𝐸) = (Base‘𝑆)
5		lanup.m	. . . 4 ⊢ 𝑀 = (𝐷 Nat 𝐸)
6	1, 5	fuchom 17923	. . 3 ⊢ 𝑀 = (Hom ‘(𝐷 FuncCat 𝐸))
7		lanup.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐸)
8	3, 7	fuchom 17923	. . 3 ⊢ 𝑁 = (Hom ‘𝑆)
9		lanup.x	. . 3 ⊢ ∙ = (comp‘𝑆)
10		ranup.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ((𝐿 ∘func 𝐹)𝑁𝑋))
11	7	natrcl 17912	. . . . 5 ⊢ (𝐴 ∈ ((𝐿 ∘func 𝐹)𝑁𝑋) → ((𝐿 ∘func 𝐹) ∈ (𝐶 Func 𝐸) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → ((𝐿 ∘func 𝐹) ∈ (𝐶 Func 𝐸) ∧ 𝑋 ∈ (𝐶 Func 𝐸)))
13	12	simprd 496	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
14	13	func1st2nd 49574	. . . . 5 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
15	14	funcrcl3 49578	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
16		lanup.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
17		opex 5404	. . . . . . 7 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → ⟨𝐷, 𝐸⟩ ∈ V)
19	16, 18	prcofelvv 49878	. . . . 5 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (V × V))
20		1st2nd2 7971	. . . . 5 ⊢ ((⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (V × V) → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
21	19, 20	syl 17	. . . 4 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = ⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩)
22	1, 15, 3, 16, 21	prcoffunca2 49885	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))((𝐷 FuncCat 𝐸) Func 𝑆)(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
23		lanup.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
24		eqidd 2740	. . . . . 6 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
25	23, 24	prcof1 49886	. . . . 5 ⊢ (𝜑 → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿) = (𝐿 ∘func 𝐹))
26	25	oveq1d 7372	. . . 4 ⊢ (𝜑 → (((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)𝑁𝑋) = ((𝐿 ∘func 𝐹)𝑁𝑋))
27	10, 26	eleqtrrd 2842	. . 3 ⊢ (𝜑 → 𝐴 ∈ (((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)𝑁𝑋))
28		eqid 2739	. . 3 ⊢ (oppCat‘(𝐷 FuncCat 𝐸)) = (oppCat‘(𝐷 FuncCat 𝐸))
29		eqid 2739	. . 3 ⊢ (oppCat‘𝑆) = (oppCat‘𝑆)
30	2, 4, 6, 8, 9, 13, 22, 23, 27, 28, 29	oppcup 49705	. 2 ⊢ (𝜑 → (𝐿(⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩((oppCat‘(𝐷 FuncCat 𝐸)) UP (oppCat‘𝑆))𝑋)𝐴 ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙)𝑁𝑋)∃!𝑏 ∈ (𝑙𝑀𝐿)𝑎 = (𝐴(⟨((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙), ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ 𝑋)((𝑙(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝐿)‘𝑏))))
31	3	fveq2i 6831	. . . 4 ⊢ (oppCat‘𝑆) = (oppCat‘(𝐶 FuncCat 𝐸))
32	28, 31, 21, 16	ranval2 50128	. . 3 ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) = (⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩((oppCat‘(𝐷 FuncCat 𝐸)) UP (oppCat‘𝑆))𝑋))
33	32	breqd 5084	. 2 ⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴 ↔ 𝐿(⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), tpos (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩((oppCat‘(𝐷 FuncCat 𝐸)) UP (oppCat‘𝑆))𝑋)𝐴))
34		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → 𝑙 ∈ (𝐷 Func 𝐸))
35		eqidd 2740	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
36	34, 35	prcof1 49886	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙) = (𝑙 ∘func 𝐹))
37	36	eqcomd 2745	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (𝑙 ∘func 𝐹) = ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))
38	37	oveq1d 7372	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → ((𝑙 ∘func 𝐹)𝑁𝑋) = (((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙)𝑁𝑋))
39	36	ad2antrr 732	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙) = (𝑙 ∘func 𝐹))
40	25	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿) = (𝐿 ∘func 𝐹))
41	39, 40	opeq12d 4813	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → ⟨((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙), ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ = ⟨(𝑙 ∘func 𝐹), (𝐿 ∘func 𝐹)⟩)
42	41	oveq1d 7372	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → (⟨((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙), ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ 𝑋) = (⟨(𝑙 ∘func 𝐹), (𝐿 ∘func 𝐹)⟩ ∙ 𝑋))
43		eqidd 2740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → 𝐴 = 𝐴)
44		simpr 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → 𝑏 ∈ (𝑙𝑀𝐿))
45		eqidd 2740	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
46	16	ad3antrrr 736	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → 𝐹 ∈ (𝐶 Func 𝐷))
47	5, 44, 45, 46	prcof21a 49889	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → ((𝑙(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝐿)‘𝑏) = (𝑏 ∘ (1st ‘𝐹)))
48	42, 43, 47	oveq123d 7378	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → (𝐴(⟨((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙), ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ 𝑋)((𝑙(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝐿)‘𝑏)) = (𝐴(⟨(𝑙 ∘func 𝐹), (𝐿 ∘func 𝐹)⟩ ∙ 𝑋)(𝑏 ∘ (1st ‘𝐹))))
49	48	eqcomd 2745	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → (𝐴(⟨(𝑙 ∘func 𝐹), (𝐿 ∘func 𝐹)⟩ ∙ 𝑋)(𝑏 ∘ (1st ‘𝐹))) = (𝐴(⟨((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙), ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ 𝑋)((𝑙(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝐿)‘𝑏)))
50	49	eqeq2d 2750	. . . . 5 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → (𝑎 = (𝐴(⟨(𝑙 ∘func 𝐹), (𝐿 ∘func 𝐹)⟩ ∙ 𝑋)(𝑏 ∘ (1st ‘𝐹))) ↔ 𝑎 = (𝐴(⟨((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙), ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ 𝑋)((𝑙(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝐿)‘𝑏))))
51	50	reubidva 3358	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) → (∃!𝑏 ∈ (𝑙𝑀𝐿)𝑎 = (𝐴(⟨(𝑙 ∘func 𝐹), (𝐿 ∘func 𝐹)⟩ ∙ 𝑋)(𝑏 ∘ (1st ‘𝐹))) ↔ ∃!𝑏 ∈ (𝑙𝑀𝐿)𝑎 = (𝐴(⟨((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙), ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ 𝑋)((𝑙(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝐿)‘𝑏))))
52	38, 51	raleqbidva 3303	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (∀𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)∃!𝑏 ∈ (𝑙𝑀𝐿)𝑎 = (𝐴(⟨(𝑙 ∘func 𝐹), (𝐿 ∘func 𝐹)⟩ ∙ 𝑋)(𝑏 ∘ (1st ‘𝐹))) ↔ ∀𝑎 ∈ (((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙)𝑁𝑋)∃!𝑏 ∈ (𝑙𝑀𝐿)𝑎 = (𝐴(⟨((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙), ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ 𝑋)((𝑙(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝐿)‘𝑏))))
53	52	ralbidva 3160	. 2 ⊢ (𝜑 → (∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)∃!𝑏 ∈ (𝑙𝑀𝐿)𝑎 = (𝐴(⟨(𝑙 ∘func 𝐹), (𝐿 ∘func 𝐹)⟩ ∙ 𝑋)(𝑏 ∘ (1st ‘𝐹))) ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙)𝑁𝑋)∃!𝑏 ∈ (𝑙𝑀𝐿)𝑎 = (𝐴(⟨((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙), ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ 𝑋)((𝑙(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝐿)‘𝑏))))
54	30, 33, 53	3bitr4d 312	1 ⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋)𝐴 ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)∃!𝑏 ∈ (𝑙𝑀𝐿)𝑎 = (𝐴(⟨(𝑙 ∘func 𝐹), (𝐿 ∘func 𝐹)⟩ ∙ 𝑋)(𝑏 ∘ (1st ‘𝐹)))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃!wreu 3342  Vcvv 3431  ⟨cop 4562   class class class wbr 5073   × cxp 5617   ∘ ccom 5623  ‘cfv 6486  (class class class)co 7357  1^st c1st 7930  2^nd c2nd 7931  tpos ctpos 8166  compcco 17224  oppCatcoppc 17669   Func cfunc 17813   ∘_func ccofu 17815   Nat cnat 17903   FuncCat cfuc 17904   UP cup 49671   −∘_F cprcof 49871   Ran cran 50104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-tpos 8167  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-ixp 8837  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-2 12236  df-3 12237  df-4 12238  df-5 12239  df-6 12240  df-7 12241  df-8 12242  df-9 12243  df-n0 12430  df-z 12517  df-dec 12637  df-uz 12781  df-fz 13454  df-struct 17109  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17172  df-hom 17236  df-cco 17237  df-cat 17626  df-cid 17627  df-oppc 17670  df-func 17817  df-cofu 17819  df-nat 17905  df-fuc 17906  df-xpc 18130  df-curf 18172  df-oppf 49621  df-up 49672  df-swapf 49758  df-fuco 49815  df-prcof 49872  df-ran 50106

This theorem is referenced by: (None)