fucofulem2 - Mathbox for Zhi Wang

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Theorem fucofulem2 48966

Description: Lemma for proving functor theorems. Maybe consider eufnfv 7232 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 2734	. . . . 5 ⊢ ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)) = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
3		eqid 2734	. . . . . 6 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
4	3	fucbas 17980	. . . . 5 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
5		eqid 2734	. . . . . 6 ⊢ (𝐶 FuncCat 𝐷) = (𝐶 FuncCat 𝐷)
6	5	fucbas 17980	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘(𝐶 FuncCat 𝐷))
7	2, 4, 6	xpcbas 18194	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)))
8	1, 7	eqtri 2757	. . 3 ⊢ 𝐵 = (Base‘((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)))
9	8	funcf2lem2 48868	. 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 7547	. . 3 ⊢ (𝐺 Fn (𝐵 × 𝐵) ↔ 𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)))
11		ffnfv 7120	. . . . . . 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 2734	. . . . . . . . . . . 12 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
13	3, 12	fuchom 17981	. . . . . . . . . . 11 ⊢ (𝐷 Nat 𝐸) = (Hom ‘(𝐷 FuncCat 𝐸))
14		eqid 2734	. . . . . . . . . . . 12 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
15	5, 14	fuchom 17981	. . . . . . . . . . 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 18196	. . . . . . . . . 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 6643	. . . . . . . . 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 7547	. . . . . . . . 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 3310	. . . . . . . . 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 6887	. . . . . . . . . . . 12 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → ((𝑚𝐺𝑛)‘𝑟) = ((𝑚𝐺𝑛)‘⟨𝑝, 𝑞⟩))
25		df-ov 7417	. . . . . . . . . . . 12 ⊢ (𝑝(𝑚𝐺𝑛)𝑞) = ((𝑚𝐺𝑛)‘⟨𝑝, 𝑞⟩)
26	24, 25	eqtr4di 2787	. . . . . . . . . . 11 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → ((𝑚𝐺𝑛)‘𝑟) = (𝑝(𝑚𝐺𝑛)𝑞))
27	26	eleq1d 2818	. . . . . . . . . 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 5834	. . . . . . . . 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 3206	. . . 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 2107 ∀wral 3050 ⟨cop 4614 × cxp 5665 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 1^st c1st 7995 2^nd c2nd 7996 ↑_m cmap 8849 Xcixp 8920 Basecbs 17230 Hom chom 17285 Func cfunc 17871 Nat cnat 17961 FuncCat cfuc 17962 ×_c cxpc 18184

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 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5261 ax-sep 5278 ax-nul 5288 ax-pow 5347 ax-pr 5414 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3773 df-csb 3882 df-dif 3936 df-un 3938 df-in 3940 df-ss 3950 df-pss 3953 df-nul 4316 df-if 4508 df-pw 4584 df-sn 4609 df-pr 4611 df-tp 4613 df-op 4615 df-uni 4890 df-iun 4975 df-br 5126 df-opab 5188 df-mpt 5208 df-tr 5242 df-id 5560 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6303 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7871 df-1st 7997 df-2nd 7998 df-frecs 8289 df-wrecs 8320 df-recs 8394 df-rdg 8433 df-1o 8489 df-er 8728 df-map 8851 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11477 df-neg 11478 df-nn 12250 df-2 12312 df-3 12313 df-4 12314 df-5 12315 df-6 12316 df-7 12317 df-8 12318 df-9 12319 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12862 df-fz 13531 df-struct 17167 df-slot 17202 df-ndx 17214 df-base 17231 df-hom 17298 df-cco 17299 df-func 17875 df-nat 17963 df-fuc 17964 df-xpc 18188

This theorem is referenced by: (None)

W3C validator