Definition df-fmla 32592

Description: Define the predicate which defines the set of valid Godel formulas. The parameter 𝑛 defines the maximum height of the formulas: the set (Fmla‘∅) is all formulas of the form 𝑥 ∈ 𝑦 (which in our coding scheme is the set ({∅} × (ω × ω)); see df-sat 32590 for the full coding scheme), see fmla0 32629, and each extra level adds to the complexity of the formulas in (Fmla‘𝑛), see fmlasuc 32633. Remark: it is sufficient to have atomic formulas of the form 𝑥 ∈ 𝑦 only, because equations (formulas of the form 𝑥 = 𝑦), which are required as (atomic) formulas, can be introduced as a defined notion in terms of ∈_𝑔, see df-goeq 32691. (Fmla‘ω) = ∪ 𝑛 ∈ ω(Fmla‘𝑛) is the set of all valid formulas, see fmla 32628. (Contributed by Mario Carneiro, 14-Jul-2013.)

Assertion

Ref	Expression
df-fmla	⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))

Detailed syntax breakdown of Definition df-fmla

Step	Hyp	Ref	Expression
1		cfmla 32584	. 2 class Fmla
2		vn	. . 3 setvar 𝑛
3		com 7580	. . . 4 class ω
4	3	csuc 6193	. . 3 class suc ω
5	2	cv 1536	. . . . 5 class 𝑛
6		c0 4291	. . . . . 6 class ∅
7		csat 32583	. . . . . 6 class Sat
8	6, 6, 7	co 7156	. . . . 5 class (∅ Sat ∅)
9	5, 8	cfv 6355	. . . 4 class ((∅ Sat ∅)‘𝑛)
10	9	cdm 5555	. . 3 class dom ((∅ Sat ∅)‘𝑛)
11	2, 4, 10	cmpt 5146	. 2 class (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))
12	1, 11	wceq 1537	1 wff Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))

This definition is referenced by:  fmlafv  32627  fmla  32628  fmlasuc0  32631