Theorem ranup 50264

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 2763	. . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2	1	fucbas 17997	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
3		lanup.s	. . . 4 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
4	3	fucbas 17997	. . 3 ⊢ (𝐶 Func 𝐸) = (Base‘𝑆)
5		lanup.m	. . . 4 ⊢ 𝑀 = (𝐷 Nat 𝐸)
6	1, 5	fuchom 17998	. . 3 ⊢ 𝑀 = (Hom ‘(𝐷 FuncCat 𝐸))
7		lanup.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐸)
8	3, 7	fuchom 17998	. . 3 ⊢ 𝑁 = (Hom ‘𝑆)
9		lanup.x	. . 3 ⊢ ∙ = (comp‘𝑆)
10		ranup.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ((𝐿 ∘func 𝐹)𝑁𝑋))
11	7	natrcl 17987	. . . . 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 499	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
14	13	func1st2nd 49698	. . . . 5 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
15	14	funcrcl3 49702	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
16		lanup.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
17		opex 5432	. . . . . . 7 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝜑 → ⟨𝐷, 𝐸⟩ ∈ V)
19	16, 18	prcofelvv 50002	. . . . 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 50009	. . 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 2764	. . . . . 6 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
25	23, 24	prcof1 50010	. . . . 5 ⊢ (𝜑 → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿) = (𝐿 ∘func 𝐹))
26	25	oveq1d 7412	. . . 4 ⊢ (𝜑 → (((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)𝑁𝑋) = ((𝐿 ∘func 𝐹)𝑁𝑋))
27	10, 26	eleqtrrd 2866	. . 3 ⊢ (𝜑 → 𝐴 ∈ (((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)𝑁𝑋))
28		eqid 2763	. . 3 ⊢ (oppCat‘(𝐷 FuncCat 𝐸)) = (oppCat‘(𝐷 FuncCat 𝐸))
29		eqid 2763	. . 3 ⊢ (oppCat‘𝑆) = (oppCat‘𝑆)
30	2, 4, 6, 8, 9, 13, 22, 23, 27, 28, 29	oppcup 49829	. 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 6871	. . . 4 ⊢ (oppCat‘𝑆) = (oppCat‘(𝐶 FuncCat 𝐸))
32	28, 31, 21, 16	ranval2 50252	. . 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 5112	. 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 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → 𝑙 ∈ (𝐷 Func 𝐸))
35		eqidd 2764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
36	34, 35	prcof1 50010	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙) = (𝑙 ∘func 𝐹))
37	36	eqcomd 2769	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (𝑙 ∘func 𝐹) = ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))
38	37	oveq1d 7412	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → ((𝑙 ∘func 𝐹)𝑁𝑋) = (((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙)𝑁𝑋))
39	36	ad2antrr 736	. . . . . . . . . 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 740	. . . . . . . . . 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 4840	. . . . . . . . 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 7412	. . . . . . . 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 2764	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → 𝐴 = 𝐴)
44		simpr 488	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → 𝑏 ∈ (𝑙𝑀𝐿))
45		eqidd 2764	. . . . . . . . 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 740	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ ((𝑙 ∘func 𝐹)𝑁𝑋)) ∧ 𝑏 ∈ (𝑙𝑀𝐿)) → 𝐹 ∈ (𝐶 Func 𝐷))
47	5, 44, 45, 46	prcof21a 50013	. . . . . . . 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 7418	. . . . . . 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 2769	. . . . . 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 2774	. . . . 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 3382	. . . 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 3327	. . 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 3184	. 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 313	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 208   ∧ wa 399   = wceq 1561   ∈ wcel 2143  ∀wral 3077  ∃!wreu 3366  Vcvv 3455  ⟨cop 4589   class class class wbr 5101   × cxp 5646   ∘ ccom 5652  ‘cfv 6522  (class class class)co 7397  1^st c1st 7969  2^nd c2nd 7970  tpos ctpos 8206  compcco 17299  oppCatcoppc 17744   Func cfunc 17888   ∘_func ccofu 17890   Nat cnat 17978   FuncCat cfuc 17979   UP cup 49795   −∘_F cprcof 49995   Ran cran 50228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  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 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-1st 7971  df-2nd 7972  df-tpos 8207  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-er 8679  df-map 8811  df-ixp 8881  df-en 8929  df-dom 8930  df-sdom 8931  df-fin 8932  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-nn 12212  df-2 12281  df-3 12282  df-4 12283  df-5 12284  df-6 12285  df-7 12286  df-8 12287  df-9 12288  df-n0 12483  df-z 12570  df-dec 12690  df-uz 12841  df-fz 13514  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17247  df-hom 17311  df-cco 17312  df-cat 17701  df-cid 17702  df-oppc 17745  df-func 17892  df-cofu 17894  df-nat 17980  df-fuc 17981  df-xpc 18205  df-curf 18247  df-oppf 49745  df-up 49796  df-swapf 49882  df-fuco 49939  df-prcof 49996  df-ran 50230

This theorem is referenced by: (None)