Definition df-fm 23886

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 23881	. 2 class FilMap
2		vx	. . 3 setvar 𝑥
3		vf	. . 3 setvar 𝑓
4		cvv 3441	. . 3 class V
5		vy	. . . 4 setvar 𝑦
6	3	cv 1541	. . . . . 6 class 𝑓
7	6	cdm 5625	. . . . 5 class dom 𝑓
8		cfbas 21301	. . . . 5 class fBas
9	7, 8	cfv 6493	. . . 4 class (fBas‘dom 𝑓)
10	2	cv 1541	. . . . 5 class 𝑥
11		vt	. . . . . . 7 setvar 𝑡
12	5	cv 1541	. . . . . . 7 class 𝑦
13	11	cv 1541	. . . . . . . 8 class 𝑡
14	6, 13	cima 5628	. . . . . . 7 class (𝑓 “ 𝑡)
15	11, 12, 14	cmpt 5180	. . . . . 6 class (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))
16	15	crn 5626	. . . . 5 class ran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))
17		cfg 21302	. . . . 5 class filGen
18	10, 16, 17	co 7360	. . . 4 class (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))
19	5, 9, 18	cmpt 5180	. . 3 class (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))))
20	2, 3, 4, 4, 19	cmpo 7362	. 2 class (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))
21	1, 20	wceq 1542	1 wff FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))

This definition is referenced by:  fmval  23891  fmf  23893