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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-fmla | Structured version Visualization version GIF version |
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 32583 for the full coding scheme), see fmla0 32622, and each extra level adds to the complexity of the formulas in (Fmla‘𝑛), see fmlasuc 32626. 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 32684. (Fmla‘ω) = ∪ 𝑛 ∈ ω(Fmla‘𝑛) is the set of all valid formulas, see fmla 32621. (Contributed by Mario Carneiro, 14-Jul-2013.) |
Ref | Expression |
---|---|
df-fmla | ⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfmla 32577 | . 2 class Fmla | |
2 | vn | . . 3 setvar 𝑛 | |
3 | com 7572 | . . . 4 class ω | |
4 | 3 | csuc 6186 | . . 3 class suc ω |
5 | 2 | cv 1530 | . . . . 5 class 𝑛 |
6 | c0 4289 | . . . . . 6 class ∅ | |
7 | csat 32576 | . . . . . 6 class Sat | |
8 | 6, 6, 7 | co 7148 | . . . . 5 class (∅ Sat ∅) |
9 | 5, 8 | cfv 6348 | . . . 4 class ((∅ Sat ∅)‘𝑛) |
10 | 9 | cdm 5548 | . . 3 class dom ((∅ Sat ∅)‘𝑛) |
11 | 2, 4, 10 | cmpt 5137 | . 2 class (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛)) |
12 | 1, 11 | wceq 1531 | 1 wff Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛)) |
Colors of variables: wff setvar class |
This definition is referenced by: fmlafv 32620 fmla 32621 fmlasuc0 32624 |
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