Metamath Proof Explorer


Theorem orvcoel

Description: If the relation produces open sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017)

Ref Expression
Hypotheses orvccel.1
|- ( ph -> S e. U. ran sigAlgebra )
orvccel.2
|- ( ph -> J e. Top )
orvccel.3
|- ( ph -> X e. ( S MblFnM ( sigaGen ` J ) ) )
orvccel.4
|- ( ph -> A e. V )
orvcoel.5
|- ( ph -> { y e. U. J | y R A } e. J )
Assertion orvcoel
|- ( ph -> ( X oRVC R A ) e. S )

Proof

Step Hyp Ref Expression
1 orvccel.1
 |-  ( ph -> S e. U. ran sigAlgebra )
2 orvccel.2
 |-  ( ph -> J e. Top )
3 orvccel.3
 |-  ( ph -> X e. ( S MblFnM ( sigaGen ` J ) ) )
4 orvccel.4
 |-  ( ph -> A e. V )
5 orvcoel.5
 |-  ( ph -> { y e. U. J | y R A } e. J )
6 1 2 3 4 orvcval4
 |-  ( ph -> ( X oRVC R A ) = ( `' X " { y e. U. J | y R A } ) )
7 2 sgsiga
 |-  ( ph -> ( sigaGen ` J ) e. U. ran sigAlgebra )
8 sssigagen
 |-  ( J e. Top -> J C_ ( sigaGen ` J ) )
9 2 8 syl
 |-  ( ph -> J C_ ( sigaGen ` J ) )
10 9 5 sseldd
 |-  ( ph -> { y e. U. J | y R A } e. ( sigaGen ` J ) )
11 1 7 3 10 mbfmcnvima
 |-  ( ph -> ( `' X " { y e. U. J | y R A } ) e. S )
12 6 11 eqeltrd
 |-  ( ph -> ( X oRVC R A ) e. S )