Theorem ranup 49621

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 2730	. . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2	1	fucbas 17931	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
3		lanup.s	. . . 4 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
4	3	fucbas 17931	. . 3 ⊢ (𝐶 Func 𝐸) = (Base‘𝑆)
5		lanup.m	. . . 4 ⊢ 𝑀 = (𝐷 Nat 𝐸)
6	1, 5	fuchom 17932	. . 3 ⊢ 𝑀 = (Hom ‘(𝐷 FuncCat 𝐸))
7		lanup.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐸)
8	3, 7	fuchom 17932	. . 3 ⊢ 𝑁 = (Hom ‘𝑆)
9		lanup.x	. . 3 ⊢ ∙ = (comp‘𝑆)
10		ranup.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ((𝐿 ∘func 𝐹)𝑁𝑋))
11	7	natrcl 17921	. . . . 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 495	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
14	13	func1st2nd 49055	. . . . 5 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
15	14	funcrcl3 49059	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
16		lanup.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
17		opex 5426	. . . . . . 7 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → ⟨𝐷, 𝐸⟩ ∈ V)
19	16, 18	prcofelvv 49359	. . . . 5 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (V × V))
20		1st2nd2 8009	. . . . 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 49366	. . 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 2731	. . . . . 6 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
25	23, 24	prcof1 49367	. . . . 5 ⊢ (𝜑 → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿) = (𝐿 ∘func 𝐹))
26	25	oveq1d 7404	. . . 4 ⊢ (𝜑 → (((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)𝑁𝑋) = ((𝐿 ∘func 𝐹)𝑁𝑋))
27	10, 26	eleqtrrd 2832	. . 3 ⊢ (𝜑 → 𝐴 ∈ (((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)𝑁𝑋))
28		eqid 2730	. . 3 ⊢ (oppCat‘(𝐷 FuncCat 𝐸)) = (oppCat‘(𝐷 FuncCat 𝐸))
29		eqid 2730	. . 3 ⊢ (oppCat‘𝑆) = (oppCat‘𝑆)
30	2, 4, 6, 8, 9, 13, 22, 23, 27, 28, 29	oppcup 49186	. 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 6863	. . . 4 ⊢ (oppCat‘𝑆) = (oppCat‘(𝐶 FuncCat 𝐸))
32	28, 31, 21, 16	ranval2 49609	. . 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 5120	. 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → 𝑙 ∈ (𝐷 Func 𝐸))
35		eqidd 2731	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
36	34, 35	prcof1 49367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙) = (𝑙 ∘func 𝐹))
37	36	eqcomd 2736	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (𝑙 ∘func 𝐹) = ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))
38	37	oveq1d 7404	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → ((𝑙 ∘func 𝐹)𝑁𝑋) = (((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙)𝑁𝑋))
39	36	ad2antrr 726	. . . . . . . . . 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 730	. . . . . . . . . 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 4847	. . . . . . . . 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 7404	. . . . . . . 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 2731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → 𝐴 = 𝐴)
44		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → 𝑏 ∈ (𝑙𝑀𝐿))
45		eqidd 2731	. . . . . . . . 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 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → 𝐹 ∈ (𝐶 Func 𝐷))
47	5, 44, 45, 46	prcof21a 49370	. . . . . . . 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 7410	. . . . . . 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 2736	. . . . . 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 2741	. . . . 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 3372	. . . 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 3307	. . 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 3155	. 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 311	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃!wreu 3354  Vcvv 3450  ⟨cop 4597   class class class wbr 5109   × cxp 5638   ∘ ccom 5644  ‘cfv 6513  (class class class)co 7389  1^st c1st 7968  2^nd c2nd 7969  tpos ctpos 8206  compcco 17238  oppCatcoppc 17678   Func cfunc 17822   ∘_func ccofu 17824   Nat cnat 17912   FuncCat cfuc 17913   UP cup 49152   −∘_F cprcof 49352   Ran cran 49585

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 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

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 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-er 8673  df-map 8803  df-ixp 8873  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-3 12251  df-4 12252  df-5 12253  df-6 12254  df-7 12255  df-8 12256  df-9 12257  df-n0 12449  df-z 12536  df-dec 12656  df-uz 12800  df-fz 13475  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17186  df-hom 17250  df-cco 17251  df-cat 17635  df-cid 17636  df-oppc 17679  df-func 17826  df-cofu 17828  df-nat 17914  df-fuc 17915  df-xpc 18139  df-curf 18181  df-oppf 49102  df-up 49153  df-swapf 49239  df-fuco 49296  df-prcof 49353  df-ran 49587

This theorem is referenced by: (None)