fucofulem2 - Mathbox for Zhi Wang

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Theorem fucofulem2 50526

Description: Lemma for proving functor theorems. Maybe consider eufnfv 7235 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 2760	. . . . 5 ⊢ ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)) = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
3	2	xpcfucbas 50467	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)))
4	1, 3	eqtri 2783	. . 3 ⊢ 𝐵 = (Base‘((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)))
5	4	funcf2lem2 50289	. 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))))
6		fnov 7551	. . 3 ⊢ (𝐺 Fn (𝐵 × 𝐵) ↔ 𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)))
7		ffnfv 7119	. . . . . . 7 ⊢ ((𝑚𝐺𝑛):(𝑚𝐻𝑛)⟶((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ ((𝑚𝐺𝑛) Fn (𝑚𝐻𝑛) ∧ ∀𝑟 ∈ (𝑚𝐻𝑛)((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
8		fucofulem2.h	. . . . . . . . . . 11 ⊢ 𝐻 = (Hom ‘((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)))
9		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑚 ∈ 𝐵)
10		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → 𝑛 ∈ 𝐵)
11	2, 4, 8, 9, 10	xpcfuchom 50469	. . . . . . . . . 10 ⊢ ((𝑚 ∈ 𝐵 ∧ 𝑛 ∈ 𝐵) → (𝑚𝐻𝑛) = (((1^st ‘𝑚)(𝐷 Nat 𝐸)(1^st ‘𝑛)) × ((2^nd ‘𝑚)(𝐶 Nat 𝐷)(2^nd ‘𝑛))))
12	11	fneq2d 6633	. . . . . . . . 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)))))
13		fnov 7551	. . . . . . . . 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)))
14	12, 13	bitrdi 290	. . . . . . . 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))))
15	11	raleqdv 3319	. . . . . . . . 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))))
16		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → ((𝑚𝐺𝑛)‘𝑟) = ((𝑚𝐺𝑛)‘⟨𝑝, 𝑞⟩))
17		df-ov 7423	. . . . . . . . . . . 12 ⊢ (𝑝(𝑚𝐺𝑛)𝑞) = ((𝑚𝐺𝑛)‘⟨𝑝, 𝑞⟩)
18	16, 17	eqtr4di 2813	. . . . . . . . . . 11 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → ((𝑚𝐺𝑛)‘𝑟) = (𝑝(𝑚𝐺𝑛)𝑞))
19	18	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → (((𝑚𝐺𝑛)‘𝑟) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛)) ↔ (𝑝(𝑚𝐺𝑛)𝑞) ∈ ((𝐹‘𝑚)(𝐶 Nat 𝐸)(𝐹‘𝑛))))
20	19	ralxp 5817	. . . . . . . . 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)))
21	15, 20	bitrdi 290	. . . . . . . 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))))
22	14, 21	anbi12d 644	. . . . . . 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)))))
23	7, 22	bitrid 286	. . . . . 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)))))
24	23	adantl 487	. . . . 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)))))
25	24	2ralbidva 3224	. . . 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)))))
26	25	mptru 1576	. . 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))))
27	6, 26	anbi12i 640	. 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)))))
28	5, 27	bitri 278	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

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 = wceq 1569 ⊤wtru 1570 ∈ wcel 2144 ∀wral 3076 ⟨cop 4834 × cxp 5648 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 1^st c1st 7999 2^nd c2nd 8000 ↑_m cmap 8850 Xcixp 8928 Basecbs 17392 Hom chom 17444 Func cfunc 18034 Nat cnat 18124 FuncCat cfuc 18125 ×_c cxpc 18347

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-er 8720 df-map 8852 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-z 12697 df-dec 12818 df-uz 12969 df-fz 13643 df-struct 17330 df-slot 17365 df-ndx 17377 df-base 17393 df-hom 17457 df-cco 17458 df-func 18038 df-nat 18126 df-fuc 18127 df-xpc 18351

This theorem is used by: fucoisfunc 50580 fucoisfunc2lem2 50582

W3C validator