Theorem ranup 49635

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 17932	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
3		lanup.s	. . . 4 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
4	3	fucbas 17932	. . 3 ⊢ (𝐶 Func 𝐸) = (Base‘𝑆)
5		lanup.m	. . . 4 ⊢ 𝑀 = (𝐷 Nat 𝐸)
6	1, 5	fuchom 17933	. . 3 ⊢ 𝑀 = (Hom ‘(𝐷 FuncCat 𝐸))
7		lanup.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐸)
8	3, 7	fuchom 17933	. . 3 ⊢ 𝑁 = (Hom ‘𝑆)
9		lanup.x	. . 3 ⊢ ∙ = (comp‘𝑆)
10		ranup.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ((𝐿 ∘func 𝐹)𝑁𝑋))
11	7	natrcl 17922	. . . . 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 49069	. . . . 5 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
15	14	funcrcl3 49073	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
16		lanup.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
17		opex 5427	. . . . . . 7 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → ⟨𝐷, 𝐸⟩ ∈ V)
19	16, 18	prcofelvv 49373	. . . . 5 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ (V × V))
20		1st2nd2 8010	. . . . 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 49380	. . 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 49381	. . . . 5 ⊢ (𝜑 → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿) = (𝐿 ∘func 𝐹))
26	25	oveq1d 7405	. . . 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 49200	. 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 6864	. . . 4 ⊢ (oppCat‘𝑆) = (oppCat‘(𝐶 FuncCat 𝐸))
32	28, 31, 21, 16	ranval2 49623	. . 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 5121	. 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 49381	. . . . . 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 7405	. . . 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 4848	. . . . . . . . 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 7405	. . . . . . . 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 49384	. . . . . . . 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 7411	. . . . . . 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 4598   class class class wbr 5110   × cxp 5639   ∘ ccom 5645  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  2^nd c2nd 7970  tpos ctpos 8207  compcco 17239  oppCatcoppc 17679   Func cfunc 17823   ∘_func ccofu 17825   Nat cnat 17913   FuncCat cfuc 17914   UP cup 49166   −∘_F cprcof 49366   Ran cran 49599

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-tpos 8208  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-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-hom 17251  df-cco 17252  df-cat 17636  df-cid 17637  df-oppc 17680  df-func 17827  df-cofu 17829  df-nat 17915  df-fuc 17916  df-xpc 18140  df-curf 18182  df-oppf 49116  df-up 49167  df-swapf 49253  df-fuco 49310  df-prcof 49367  df-ran 49601

This theorem is referenced by: (None)