Definition df-fm 22546

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 22541	. 2 class FilMap
2		vx	. . 3 setvar 𝑥
3		vf	. . 3 setvar 𝑓
4		cvv 3494	. . 3 class V
5		vy	. . . 4 setvar 𝑦
6	3	cv 1536	. . . . . 6 class 𝑓
7	6	cdm 5555	. . . . 5 class dom 𝑓
8		cfbas 20533	. . . . 5 class fBas
9	7, 8	cfv 6355	. . . 4 class (fBas‘dom 𝑓)
10	2	cv 1536	. . . . 5 class 𝑥
11		vt	. . . . . . 7 setvar 𝑡
12	5	cv 1536	. . . . . . 7 class 𝑦
13	11	cv 1536	. . . . . . . 8 class 𝑡
14	6, 13	cima 5558	. . . . . . 7 class (𝑓 “ 𝑡)
15	11, 12, 14	cmpt 5146	. . . . . 6 class (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))
16	15	crn 5556	. . . . 5 class ran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))
17		cfg 20534	. . . . 5 class filGen
18	10, 16, 17	co 7156	. . . 4 class (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))
19	5, 9, 18	cmpt 5146	. . 3 class (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))))
20	2, 3, 4, 4, 19	cmpo 7158	. 2 class (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))
21	1, 20	wceq 1537	1 wff FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))

This definition is referenced by:  fmval  22551  fmf  22553