Definition df-fm 24106

Description: Define a function that takes a filter to a neighborhood filter of the range. (Since we now allow filter bases to have support smaller than the base set, the function has to come first to ensure that curryings are sets.) (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 20-Jul-2015.)

Assertion

Ref	Expression
df-fm	⊢ FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))

Distinct variable group:   𝑡,𝑓,𝑥,𝑦

Detailed syntax breakdown of Definition df-fm

Step	Hyp	Ref	Expression
1		cfm 24101	. 2 class FilMap
2		vx	. . 3 setvar 𝑥
3		vf	. . 3 setvar 𝑓
4		cvv 3454	. . 3 class V
5		vy	. . . 4 setvar 𝑦
6	3	cv 1568	. . . . . 6 class 𝑓
7	6	cdm 5660	. . . . 5 class dom 𝑓
8		cfbas 21521	. . . . 5 class fBas
9	7, 8	cfv 6536	. . . 4 class (fBas‘dom 𝑓)
10	2	cv 1568	. . . . 5 class 𝑥
11		vt	. . . . . . 7 setvar 𝑡
12	5	cv 1568	. . . . . . 7 class 𝑦
13	11	cv 1568	. . . . . . . 8 class 𝑡
14	6, 13	cima 5663	. . . . . . 7 class (𝑓 “ 𝑡)
15	11, 12, 14	cmpt 5191	. . . . . 6 class (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))
16	15	crn 5661	. . . . 5 class ran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))
17		cfg 21522	. . . . 5 class filGen
18	10, 16, 17	co 7412	. . . 4 class (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))
19	5, 9, 18	cmpt 5191	. . 3 class (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))))
20	2, 3, 4, 4, 19	cmpo 7414	. 2 class (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))
21	1, 20	wceq 1569	1 wff FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))

This definition is used by:  fmval  24111  fmf  24113