Theorem fucoisfunc 50578

Description: A characterization of functors with the object part of functor composition, in terms of an exact morphism function and compatible component functors. See fucoisfunc2 50581 for the componentwise form. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypotheses

Ref	Expression
fucoisfunc.w	⊢ 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))
fucoisfunc.t	⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
fucoisfunc.q	⊢ 𝑄 = (𝐶 FuncCat 𝐸)
fucoisfunc.o	⊢ 𝑂 = ( ∘func ↾ 𝑊)
fucoisfunc.h	⊢ 𝐻 = (Hom ‘𝑇)
fucoisfunc.i	⊢ 𝐼 = (Id‘𝑇)
fucoisfunc.j	⊢ 𝐽 = (Id‘𝑄)
fucoisfunc.x	⊢ · = (comp‘𝑇)
fucoisfunc.y	⊢ ⚬ = (comp‘𝑄)
fucoisfunc.c	⊢ (𝜑 → 𝐶 ∈ Cat)
fucoisfunc.d	⊢ (𝜑 → 𝐷 ∈ Cat)
fucoisfunc.e	⊢ (𝜑 → 𝐸 ∈ Cat)

Assertion

Ref

Expression

fucoisfunc

⊢ (𝜑 → (𝑂(𝑇 Func 𝑄)𝑃 ↔ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)))) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))))))

Distinct variable groups:   𝐶,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝐷,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝐸,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝑊,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝑂,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝑃,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝑇,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝑄,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝐻,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝐼,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝐽,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   𝜑,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   · ,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧   ⚬ ,𝑎,𝑏,𝑚,𝑛,𝑝,𝑞,𝑢,𝑣,𝑥,𝑦,𝑧

Proof of Theorem fucoisfunc

Step	Hyp	Ref	Expression
1		fucoisfunc.w	. . . . 5 ⊢ 𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))
2		fucoisfunc.t	. . . . . 6 ⊢ 𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))
3	2	xpcfucbas 50465	. . . . 5 ⊢ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)) = (Base‘𝑇)
4	1, 3	eqtri 2783	. . . 4 ⊢ 𝑊 = (Base‘𝑇)
5		fucoisfunc.q	. . . . 5 ⊢ 𝑄 = (𝐶 FuncCat 𝐸)
6	5	fucbas 18140	. . . 4 ⊢ (𝐶 Func 𝐸) = (Base‘𝑄)
7		fucoisfunc.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝑇)
8		eqid 2760	. . . . 5 ⊢ (𝐶 Nat 𝐸) = (𝐶 Nat 𝐸)
9	5, 8	fuchom 18141	. . . 4 ⊢ (𝐶 Nat 𝐸) = (Hom ‘𝑄)
10		fucoisfunc.i	. . . 4 ⊢ 𝐼 = (Id‘𝑇)
11		fucoisfunc.j	. . . 4 ⊢ 𝐽 = (Id‘𝑄)
12		fucoisfunc.x	. . . 4 ⊢ · = (comp‘𝑇)
13		fucoisfunc.y	. . . 4 ⊢ ⚬ = (comp‘𝑄)
14		eqid 2760	. . . . . 6 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
15		fucoisfunc.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
16		fucoisfunc.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
17	14, 15, 16	fuccat 18150	. . . . 5 ⊢ (𝜑 → (𝐷 FuncCat 𝐸) ∈ Cat)
18		eqid 2760	. . . . . 6 ⊢ (𝐶 FuncCat 𝐷) = (𝐶 FuncCat 𝐷)
19		fucoisfunc.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
20	18, 19, 15	fuccat 18150	. . . . 5 ⊢ (𝜑 → (𝐶 FuncCat 𝐷) ∈ Cat)
21	2, 17, 20	xpccat 18366	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Cat)
22	5, 19, 16	fuccat 18150	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Cat)
23	4, 6, 7, 9, 10, 11, 12, 13, 21, 22	isfunc 18041	. . 3 ⊢ (𝜑 → (𝑂(𝑇 Func 𝑄)𝑃 ↔ (𝑂:𝑊⟶(𝐶 Func 𝐸) ∧ 𝑃 ∈ X𝑧 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))))))
24		rescofuf 50299	. . . . 5 ⊢ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))):((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸)
25		fucoisfunc.o	. . . . . . 7 ⊢ 𝑂 = ( ∘func ↾ 𝑊)
26	1	reseq2i 5966	. . . . . . 7 ⊢ ( ∘func ↾ 𝑊) = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
27	25, 26	eqtri 2783	. . . . . 6 ⊢ 𝑂 = ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷)))
28	27, 1	feq12i 6702	. . . . 5 ⊢ (𝑂:𝑊⟶(𝐶 Func 𝐸) ↔ ( ∘func ↾ ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))):((𝐷 Func 𝐸) × (𝐶 Func 𝐷))⟶(𝐶 Func 𝐸))
29	24, 28	mpbir 234	. . . 4 ⊢ 𝑂:𝑊⟶(𝐶 Func 𝐸)
30		3anass 1111	. . . 4 ⊢ ((𝑂:𝑊⟶(𝐶 Func 𝐸) ∧ 𝑃 ∈ X𝑧 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)))) ↔ (𝑂:𝑊⟶(𝐶 Func 𝐸) ∧ (𝑃 ∈ X𝑧 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))))))
31	29, 30	mpbiran 722	. . 3 ⊢ ((𝑂:𝑊⟶(𝐶 Func 𝐸) ∧ 𝑃 ∈ X𝑧 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)))) ↔ (𝑃 ∈ X𝑧 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)))))
32	23, 31	bitrdi 290	. 2 ⊢ (𝜑 → (𝑂(𝑇 Func 𝑄)𝑃 ↔ (𝑃 ∈ X𝑧 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))))))
33	2	fveq2i 6888	. . . . 5 ⊢ (Hom ‘𝑇) = (Hom ‘((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷)))
34	7, 33	eqtri 2783	. . . 4 ⊢ 𝐻 = (Hom ‘((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷)))
35	1, 34	fucofulem2 50524	. . 3 ⊢ (𝑃 ∈ X𝑧 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)))))
36	35	anbi1i 636	. 2 ⊢ ((𝑃 ∈ X𝑧 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑧))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)))) ↔ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)))) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚)))))
37	32, 36	bitrdi 290	1 ⊢ (𝜑 → (𝑂(𝑇 Func 𝑄)𝑃 ↔ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)))) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ⟨cop 4589   class class class wbr 5102   × cxp 5648   ↾ cres 5652  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000   ↑_m cmap 8849  Xcixp 8927  Basecbs 17389  Hom chom 17441  compcco 17442  Catccat 17840  Idccid 17841   Func cfunc 18031   ∘_func ccofu 18033   Nat cnat 18121   FuncCat cfuc 18122   ×_c cxpc 18344

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 11258  ax-resscn 11259  ax-1cn 11260  ax-icn 11261  ax-addcl 11262  ax-addrcl 11263  ax-mulcl 11264  ax-mulrcl 11265  ax-mulcom 11266  ax-addass 11267  ax-mulass 11268  ax-distr 11269  ax-i2m1 11270  ax-1ne0 11271  ax-1rid 11272  ax-rnegex 11273  ax-rrecex 11274  ax-cnre 11275  ax-pre-lttri 11276  ax-pre-lttrn 11277  ax-pre-ltadd 11278  ax-pre-mulgt0 11279

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-rmo 3365  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 4584  df-pr 4586  df-tp 4588  df-op 4590  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 8301  df-wrecs 8332  df-recs 8381  df-rdg 8420  df-1o 8478  df-er 8719  df-map 8851  df-ixp 8928  df-en 8976  df-dom 8977  df-sdom 8978  df-fin 8979  df-pnf 11347  df-mnf 11348  df-xr 11349  df-ltxr 11350  df-le 11351  df-sub 11545  df-neg 11546  df-nn 12338  df-2 12407  df-3 12408  df-4 12409  df-5 12410  df-6 12411  df-7 12412  df-8 12413  df-9 12414  df-n0 12609  df-z 12696  df-dec 12817  df-uz 12968  df-fz 13642  df-struct 17327  df-slot 17362  df-ndx 17374  df-base 17390  df-hom 17454  df-cco 17455  df-cat 17844  df-cid 17845  df-func 18035  df-cofu 18037  df-nat 18123  df-fuc 18124  df-xpc 18348

This theorem is used by:  fucoisfunc2lem2  50580