Metamath Proof Explorer

Theorem nsssmfmbflem

Description: The sigma-measurable functions (w.r.t. the Lebesgue measure on the Reals) are not a subset of the measurable functions. (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Hypotheses nsssmfmbflem.s 
|- S = dom vol
nsssmfmbflem.x 
|- ( ph -> X C_ RR )
nsssmfmbflem.n 
|- ( ph -> -. X e. S )
nsssmfmbflem.f 
|- F = ( x e. X |-> 0 )
Assertion nsssmfmbflem 
|- ( ph -> E. f ( f e. ( SMblFn ` S ) /\ -. f e. MblFn ) )

Proof

Step Hyp Ref Expression
1 nsssmfmbflem.s 
 |-  S = dom vol
2 nsssmfmbflem.x 
 |-  ( ph -> X C_ RR )
3 nsssmfmbflem.n 
 |-  ( ph -> -. X e. S )
4 nsssmfmbflem.f 
 |-  F = ( x e. X |-> 0 )
5 0red 
 |-  ( ( ph /\ x e. X ) -> 0 e. RR )
6 5 4 fmptd 
 |-  ( ph -> F : X --> RR )
7 reex 
 |-  RR e. _V
8 7 a1i 
 |-  ( ph -> RR e. _V )
9 8 2 ssexd 
 |-  ( ph -> X e. _V )
10 6 9 fexd 
 |-  ( ph -> F e. _V )
11 1 2 3 4 smfmbfcex 
 |-  ( ph -> ( F e. ( SMblFn ` S ) /\ -. F e. MblFn ) )
12 eleq1 
 |-  ( f = F -> ( f e. ( SMblFn ` S ) <-> F e. ( SMblFn ` S ) ) )
13 eleq1 
 |-  ( f = F -> ( f e. MblFn <-> F e. MblFn ) )
14 13 notbid 
 |-  ( f = F -> ( -. f e. MblFn <-> -. F e. MblFn ) )
15 12 14 anbi12d 
 |-  ( f = F -> ( ( f e. ( SMblFn ` S ) /\ -. f e. MblFn ) <-> ( F e. ( SMblFn ` S ) /\ -. F e. MblFn ) ) )
16 15 spcegv 
 |-  ( F e. _V -> ( ( F e. ( SMblFn ` S ) /\ -. F e. MblFn ) -> E. f ( f e. ( SMblFn ` S ) /\ -. f e. MblFn ) ) )
17 10 11 16 sylc 
 |-  ( ph -> E. f ( f e. ( SMblFn ` S ) /\ -. f e. MblFn ) )