Description: If Z is a result of an application of the procedure R to an object Y that follows X in the R -sequence and if every result of an application of the procedure R to X has a property A that is hereditary in the R -sequence, then Z has property A . Proposition 87 of Frege1879 p. 66. (Contributed by RP, 1-Jul-2020) (Revised by RP, 7-Jul-2020) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | frege87.x | |- X e. U |
|
| frege87.y | |- Y e. V |
||
| frege87.z | |- Z e. W |
||
| frege87.r | |- R e. S |
||
| frege87.a | |- A e. B |
||
| Assertion | frege87 | |- ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frege87.x | |- X e. U |
|
| 2 | frege87.y | |- Y e. V |
|
| 3 | frege87.z | |- Z e. W |
|
| 4 | frege87.r | |- R e. S |
|
| 5 | frege87.a | |- A e. B |
|
| 6 | 2 3 | frege73 | |- ( ( R hereditary A -> Y e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) |
| 7 | 1 2 4 5 | frege86 | |- ( ( ( R hereditary A -> Y e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) -> ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) ) ) |
| 8 | 6 7 | ax-mp | |- ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) ) |