Definition df-fm 23098

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 23093	. 2 class FilMap
2		vx	. . 3 setvar 𝑥
3		vf	. . 3 setvar 𝑓
4		cvv 3433	. . 3 class V
5		vy	. . . 4 setvar 𝑦
6	3	cv 1538	. . . . . 6 class 𝑓
7	6	cdm 5590	. . . . 5 class dom 𝑓
8		cfbas 20594	. . . . 5 class fBas
9	7, 8	cfv 6437	. . . 4 class (fBas‘dom 𝑓)
10	2	cv 1538	. . . . 5 class 𝑥
11		vt	. . . . . . 7 setvar 𝑡
12	5	cv 1538	. . . . . . 7 class 𝑦
13	11	cv 1538	. . . . . . . 8 class 𝑡
14	6, 13	cima 5593	. . . . . . 7 class (𝑓 “ 𝑡)
15	11, 12, 14	cmpt 5158	. . . . . 6 class (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))
16	15	crn 5591	. . . . 5 class ran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))
17		cfg 20595	. . . . 5 class filGen
18	10, 16, 17	co 7284	. . . 4 class (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))
19	5, 9, 18	cmpt 5158	. . 3 class (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))))
20	2, 3, 4, 4, 19	cmpo 7286	. 2 class (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))
21	1, 20	wceq 1539	1 wff FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))

This definition is referenced by:  fmval  23103  fmf  23105