Description: Given a sigma-measurable function, the preimage of a Borel set belongs to the subspace sigma-algebra induced by the domain of the function. Proposition 121E (f) of Fremlin1 p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | smfpimbor1.s | |- ( ph -> S e. SAlg ) |
|
| smfpimbor1.f | |- ( ph -> F e. ( SMblFn ` S ) ) |
||
| smfpimbor1.a | |- D = dom F |
||
| smfpimbor1.j | |- J = ( topGen ` ran (,) ) |
||
| smfpimbor1.b | |- B = ( SalGen ` J ) |
||
| smfpimbor1.e | |- ( ph -> E e. B ) |
||
| smfpimbor1.p | |- P = ( `' F " E ) |
||
| Assertion | smfpimbor1 | |- ( ph -> P e. ( S |`t D ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | smfpimbor1.s | |- ( ph -> S e. SAlg ) |
|
| 2 | smfpimbor1.f | |- ( ph -> F e. ( SMblFn ` S ) ) |
|
| 3 | smfpimbor1.a | |- D = dom F |
|
| 4 | smfpimbor1.j | |- J = ( topGen ` ran (,) ) |
|
| 5 | smfpimbor1.b | |- B = ( SalGen ` J ) |
|
| 6 | smfpimbor1.e | |- ( ph -> E e. B ) |
|
| 7 | smfpimbor1.p | |- P = ( `' F " E ) |
|
| 8 | eqid | |- { e e. ~P RR | ( `' F " e ) e. ( S |`t D ) } = { e e. ~P RR | ( `' F " e ) e. ( S |`t D ) } |
|
| 9 | 1 2 3 4 5 6 7 8 | smfpimbor1lem2 | |- ( ph -> P e. ( S |`t D ) ) |