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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 e. y (which in our coding scheme is the set ( { (/) } X. (om X. om ) ) ; see df-sat for the full coding scheme), see fmla0 , and each extra level adds to the complexity of the formulas in ( Fmlan ) , see fmlasuc . Remark: it is sufficient to have atomic formulas of the form x e. 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 e.g , see df-goeq . ( Fmla_om ) = U_ n e. _om ( Fmlan ) is the set of all valid formulas, see fmla . (Contributed by Mario Carneiro, 14-Jul-2013)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | df-fmla | |- Fmla = ( n e. suc _om |-> dom ( ( (/) Sat (/) ) ` n ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | cfmla | |- Fmla |
|
| 1 | vn | |- n |
|
| 2 | com | |- _om |
|
| 3 | 2 | csuc | |- suc _om |
| 4 | c0 | |- (/) |
|
| 5 | csat | |- Sat |
|
| 6 | 4 4 5 | co | |- ( (/) Sat (/) ) |
| 7 | 1 | cv | |- n |
| 8 | 7 6 | cfv | |- ( ( (/) Sat (/) ) ` n ) |
| 9 | 8 | cdm | |- dom ( ( (/) Sat (/) ) ` n ) |
| 10 | 1 3 9 | cmpt | |- ( n e. suc _om |-> dom ( ( (/) Sat (/) ) ` n ) ) |
| 11 | 0 10 | wceq | |- Fmla = ( n e. suc _om |-> dom ( ( (/) Sat (/) ) ` n ) ) |