fucofulem2 - Mathbox for Zhi Wang

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Theorem fucofulem2 50648

Description: Lemma for proving functor theorems. Maybe consider eufnfv 7261 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 2763	. . . . 5 ⊢ ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)) = ((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷))
3	2	xpcfucbas 50589	. . . 4 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)))
4	1, 3	eqtri 2786	. . 3 ⊢ 𝐵 = (Base‘((𝐷 FuncCat 𝐸) ×_c (𝐶 FuncCat 𝐷)))
5	4	funcf2lem2 50411	. 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 7577	. . 3 ⊢ (𝐺 Fn (𝐵 × 𝐵) ↔ 𝐺 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑢𝐺𝑣)))
7		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))))
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 50591	. . . . . . . . . 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 6658	. . . . . . . . 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 7577	. . . . . . . . 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 3323	. . . . . . . . 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 6911	. . . . . . . . . . . 12 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → ((𝑚𝐺𝑛)‘𝑟) = ((𝑚𝐺𝑛)‘⟨𝑝, 𝑞⟩))
17		df-ov 7449	. . . . . . . . . . . 12 ⊢ (𝑝(𝑚𝐺𝑛)𝑞) = ((𝑚𝐺𝑛)‘⟨𝑝, 𝑞⟩)
18	16, 17	eqtr4di 2816	. . . . . . . . . . 11 ⊢ (𝑟 = ⟨𝑝, 𝑞⟩ → ((𝑚𝐺𝑛)‘𝑟) = (𝑝(𝑚𝐺𝑛)𝑞))
19	18	eleq1d 2849	. . . . . . . . . 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 5825	. . . . . . . . 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 3228	. . . 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 1577	. . 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 1570 ⊤wtru 1571 ∈ wcel 2147 ∀wral 3080 ⟨cop 4838 × cxp 5654 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7446 ∈ cmpo 7448 1^st c1st 8026 2^nd c2nd 8027 ↑_m cmap 8878 Xcixp 8956 Basecbs 17419 Hom chom 17471 Func cfunc 18061 Nat cnat 18151 FuncCat cfuc 18152 ×_c cxpc 18374

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-om 7905 df-1st 8028 df-2nd 8029 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-er 8748 df-map 8880 df-ixp 8957 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-nn 12366 df-2 12435 df-3 12436 df-4 12437 df-5 12438 df-6 12439 df-7 12440 df-8 12441 df-9 12442 df-n0 12637 df-z 12724 df-dec 12845 df-uz 12996 df-fz 13670 df-struct 17357 df-slot 17392 df-ndx 17404 df-base 17420 df-hom 17484 df-cco 17485 df-func 18065 df-nat 18153 df-fuc 18154 df-xpc 18378

This theorem is used by: fucoisfunc 50702 fucoisfunc2lem2 50704

W3C validator