fucofulem2 - Mathbox for Zhi Wang

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Theorem fucofulem2 48880

Description: Lemma for proving functor theorems. Maybe consider eufnfv 7256 to prove the uniqueness of a functor. (Contributed by Zhi Wang, 25-Sep-2025.)

**Hypotheses**
Ref	Expression
fucofulem2.b	⊢ 𝐵 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))
fucofulem2.h	⊢ 𝐻 = (Hom ‘((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)))

**Assertion**
Ref	Expression
fucofulem2	⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1^st ‘𝑧))(𝐶 Nat 𝐸)(𝐹‘(2^nd ‘𝑧))) ↑_m (𝐻‘𝑧)) ↔ (𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)) ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝑚𝐺𝑛) = (𝑏 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)), 𝑎 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛))∀𝑞 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))

Distinct variable groups: 𝐵,𝑚,𝑛,𝑧 𝑢,𝐵,𝑣 𝐶,𝑎,𝑏,𝑚,𝑛 𝐶,𝑝,𝑞,𝑚,𝑛 𝑧,𝐶 𝐷,𝑎,𝑏 𝐷,𝑝,𝑞 𝐸,𝑎,𝑏,𝑚,𝑛 𝐸,𝑝,𝑞 𝑧,𝐸 𝑚,𝐹,𝑛,𝑝,𝑞 𝑧,𝐹 𝐺,𝑎,𝑏,𝑚,𝑛 𝐺,𝑝,𝑞 𝑢,𝐺,𝑣 𝑧,𝐺 𝑚,𝐻,𝑛,𝑧 Allowed substitution hints: 𝐵(𝑞,𝑝,𝑎,𝑏) 𝐶(𝑣,𝑢) 𝐷(𝑧,𝑣,𝑢,𝑚,𝑛) 𝐸(𝑣,𝑢) 𝐹(𝑣,𝑢,𝑎,𝑏) 𝐻(𝑣,𝑢,𝑞,𝑝,𝑎,𝑏)

Proof of Theorem fucofulem2 Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucofulem2.b	. . . 4 ⊢ 𝐵 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))
2		eqid 2737	. . . . 5 ⊢ ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)) = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
3		eqid 2737	. . . . . 6 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
4	3	fucbas 18025	. . . . 5 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
5		eqid 2737	. . . . . 6 ⊢ (𝐶 FuncCat 𝐷) = (𝐶 FuncCat 𝐷)
6	5	fucbas 18025	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘(𝐶 FuncCat 𝐷))
7	2, 4, 6	xpcbas 18243	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)))
8	1, 7	eqtri 2765	. . 3 ⊢ 𝐵 = (Base‘((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)))
9	8	funcf2lem2 48840	. 2 ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1^st ‘𝑧))(𝐶 Nat 𝐸)(𝐹‘(2^nd ‘𝑧))) ↑_m (𝐻‘𝑧)) ↔ (𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 (𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
10		fnov 7571	. . 3 ⊢ (𝐺 Fn (𝐵 × 𝐵) ↔ 𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)))
11		ffnfv 7146	. . . . . . 7 ⊢ ((𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ((𝑚𝐺𝑛) Fn (𝑚𝐻𝑛) ∧ ∀𝑟 ∈ (𝑚𝐻𝑛)((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
12		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
13	3, 12	fuchom 18026	. . . . . . . . . . 11 ⊢ (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
14		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
15	5, 14	fuchom 18026	. . . . . . . . . . 11 ⊢ (𝐶 Nat 𝐷) = (Hom ‘(𝐶 FuncCat 𝐷))
16		fucofulem2.h	. . . . . . . . . . 11 ⊢ 𝐻 = (Hom ‘((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)))
17		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑚 ∈ 𝐵)
18		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ 𝐵)
19	2, 8, 13, 15, 16, 17, 18	xpchom 18245	. . . . . . . . . 10 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (𝑚𝐻𝑛) = (((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)) × ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛))))
20	19	fneq2d 6670	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → ((𝑚𝐺𝑛) Fn (𝑚𝐻𝑛) ↔ (𝑚𝐺𝑛) Fn (((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)) × ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛)))))
21		fnov 7571	. . . . . . . . 9 ⊢ ((𝑚𝐺𝑛) Fn (((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)) × ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛))) ↔ (𝑚𝐺𝑛) = (𝑏 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)), 𝑎 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)))
22	20, 21	bitrdi 287	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → ((𝑚𝐺𝑛) Fn (𝑚𝐻𝑛) ↔ (𝑚𝐺𝑛) = (𝑏 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)), 𝑎 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎))))
23	19	raleqdv 3326	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (∀𝑟 ∈ (𝑚𝐻𝑛)((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ∀𝑟 ∈ (((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)) × ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛)))((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
24		fveq2 6914	. . . . . . . . . . . 12 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → ((𝑚𝐺𝑛)‘𝑟) = ((𝑚𝐺𝑛)‘⟨𝑝, 𝑞⟩))
25		df-ov 7441	. . . . . . . . . . . 12 ⊢ (𝑝(𝑚𝐺𝑛)𝑞) = ((𝑚𝐺𝑛)‘⟨𝑝, 𝑞⟩)
26	24, 25	eqtr4di 2795	. . . . . . . . . . 11 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → ((𝑚𝐺𝑛)‘𝑟) = (𝑝(𝑚𝐺𝑛)𝑞))
27	26	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → (((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ (𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
28	27	ralxp 5859	. . . . . . . . 9 ⊢ (∀𝑟 ∈ (((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)) × ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛)))((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ∀𝑝 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛))∀𝑞 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))
29	23, 28	bitrdi 287	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (∀𝑟 ∈ (𝑚𝐻𝑛)((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ∀𝑝 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛))∀𝑞 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
30	22, 29	anbi12d 632	. . . . . . 7 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (((𝑚𝐺𝑛) Fn (𝑚𝐻𝑛) ∧ ∀𝑟 ∈ (𝑚𝐻𝑛)((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))) ↔ ((𝑚𝐺𝑛) = (𝑏 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)), 𝑎 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛))∀𝑞 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))
31	11, 30	bitrid 283	. . . . . 6 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → ((𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ((𝑚𝐺𝑛) = (𝑏 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)), 𝑎 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛))∀𝑞 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))
32	31	adantl 481	. . . . 5 ⊢ ((⊤ ∧ (𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵)) → ((𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ((𝑚𝐺𝑛) = (𝑏 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)), 𝑎 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛))∀𝑞 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))
33	32	2ralbidva 3219	. . . 4 ⊢ (⊤ → (∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 (𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝑚𝐺𝑛) = (𝑏 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)), 𝑎 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛))∀𝑞 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))
34	33	mptru 1546	. . 3 ⊢ (∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 (𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝑚𝐺𝑛) = (𝑏 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)), 𝑎 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛))∀𝑞 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
35	10, 34	anbi12i 628	. 2 ⊢ ((𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 (𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))) ↔ (𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)) ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝑚𝐺𝑛) = (𝑏 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)), 𝑎 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛))∀𝑞 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))
36	9, 35	bitri 275	1 ⊢ (𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1^st ‘𝑧))(𝐶 Nat 𝐸)(𝐹‘(2^nd ‘𝑧))) ↑_m (𝐻‘𝑧)) ↔ (𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)) ∧ ∀𝑚 ∈ 𝐵 ∀𝑛 ∈ 𝐵 ((𝑚𝐺𝑛) = (𝑏 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)), 𝑎 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛)) ↦ (𝑏(𝑚𝐺𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛))∀𝑞 ∈ ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛))(𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)))))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ⊤wtru 1540 ∈ wcel 2108 ∀wral 3061 ⟨cop 4640 × cxp 5691 Fn wfn 6564 ⟶wf 6565 ‘cfv 6569 (class class class)co 7438 ∈ cmpo 7440 1^st c1st 8020 2^nd c2nd 8021 ↑_m cmap 8874 Xcixp 8945 Basecbs 17254 Hom chom 17318 Func cfunc 17914 Nat cnat 18005 FuncCat cfuc 18006 ×_c cxpc 18233

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-tp 4639 df-op 4641 df-uni 4916 df-iun 5001 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-om 7895 df-1st 8022 df-2nd 8023 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-1o 8514 df-er 8753 df-map 8876 df-ixp 8946 df-en 8994 df-dom 8995 df-sdom 8996 df-fin 8997 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-nn 12274 df-2 12336 df-3 12337 df-4 12338 df-5 12339 df-6 12340 df-7 12341 df-8 12342 df-9 12343 df-n0 12534 df-z 12621 df-dec 12741 df-uz 12886 df-fz 13554 df-struct 17190 df-slot 17225 df-ndx 17237 df-base 17255 df-hom 17331 df-cco 17332 df-func 17918 df-nat 18007 df-fuc 18008 df-xpc 18237

This theorem is referenced by: (None)

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