Definition df-fm 24000

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 23995	. 2 class FilMap
2		vx	. . 3 setvar 𝑥
3		vf	. . 3 setvar 𝑓
4		cvv 3456	. . 3 class V
5		vy	. . . 4 setvar 𝑦
6	3	cv 1561	. . . . . 6 class 𝑓
7	6	cdm 5649	. . . . 5 class dom 𝑓
8		cfbas 21414	. . . . 5 class fBas
9	7, 8	cfv 6523	. . . 4 class (fBas‘dom 𝑓)
10	2	cv 1561	. . . . 5 class 𝑥
11		vt	. . . . . . 7 setvar 𝑡
12	5	cv 1561	. . . . . . 7 class 𝑦
13	11	cv 1561	. . . . . . . 8 class 𝑡
14	6, 13	cima 5652	. . . . . . 7 class (𝑓 “ 𝑡)
15	11, 12, 14	cmpt 5183	. . . . . 6 class (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))
16	15	crn 5650	. . . . 5 class ran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))
17		cfg 21415	. . . . 5 class filGen
18	10, 16, 17	co 7398	. . . 4 class (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))
19	5, 9, 18	cmpt 5183	. . 3 class (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))))
20	2, 3, 4, 4, 19	cmpo 7400	. 2 class (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))
21	1, 20	wceq 1562	1 wff FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))

This definition is referenced by:  fmval  24005  fmf  24007