Metamath Proof Explorer


Theorem sibfima

Description: Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018)

Ref Expression
Hypotheses sitgval.b 𝐵 = ( Base ‘ 𝑊 )
sitgval.j 𝐽 = ( TopOpen ‘ 𝑊 )
sitgval.s 𝑆 = ( sigaGen ‘ 𝐽 )
sitgval.0 0 = ( 0g𝑊 )
sitgval.x · = ( ·𝑠𝑊 )
sitgval.h 𝐻 = ( ℝHom ‘ ( Scalar ‘ 𝑊 ) )
sitgval.1 ( 𝜑𝑊𝑉 )
sitgval.2 ( 𝜑𝑀 ran measures )
sibfmbl.1 ( 𝜑𝐹 ∈ dom ( 𝑊 sitg 𝑀 ) )
Assertion sibfima ( ( 𝜑𝐴 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑀 ‘ ( 𝐹 “ { 𝐴 } ) ) ∈ ( 0 [,) +∞ ) )

Proof

Step Hyp Ref Expression
1 sitgval.b 𝐵 = ( Base ‘ 𝑊 )
2 sitgval.j 𝐽 = ( TopOpen ‘ 𝑊 )
3 sitgval.s 𝑆 = ( sigaGen ‘ 𝐽 )
4 sitgval.0 0 = ( 0g𝑊 )
5 sitgval.x · = ( ·𝑠𝑊 )
6 sitgval.h 𝐻 = ( ℝHom ‘ ( Scalar ‘ 𝑊 ) )
7 sitgval.1 ( 𝜑𝑊𝑉 )
8 sitgval.2 ( 𝜑𝑀 ran measures )
9 sibfmbl.1 ( 𝜑𝐹 ∈ dom ( 𝑊 sitg 𝑀 ) )
10 1 2 3 4 5 6 7 8 issibf ( 𝜑 → ( 𝐹 ∈ dom ( 𝑊 sitg 𝑀 ) ↔ ( 𝐹 ∈ ( dom 𝑀 MblFnM 𝑆 ) ∧ ran 𝐹 ∈ Fin ∧ ∀ 𝑥 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑀 ‘ ( 𝐹 “ { 𝑥 } ) ) ∈ ( 0 [,) +∞ ) ) ) )
11 9 10 mpbid ( 𝜑 → ( 𝐹 ∈ ( dom 𝑀 MblFnM 𝑆 ) ∧ ran 𝐹 ∈ Fin ∧ ∀ 𝑥 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑀 ‘ ( 𝐹 “ { 𝑥 } ) ) ∈ ( 0 [,) +∞ ) ) )
12 11 simp3d ( 𝜑 → ∀ 𝑥 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑀 ‘ ( 𝐹 “ { 𝑥 } ) ) ∈ ( 0 [,) +∞ ) )
13 sneq ( 𝑥 = 𝐴 → { 𝑥 } = { 𝐴 } )
14 13 imaeq2d ( 𝑥 = 𝐴 → ( 𝐹 “ { 𝑥 } ) = ( 𝐹 “ { 𝐴 } ) )
15 14 fveq2d ( 𝑥 = 𝐴 → ( 𝑀 ‘ ( 𝐹 “ { 𝑥 } ) ) = ( 𝑀 ‘ ( 𝐹 “ { 𝐴 } ) ) )
16 15 eleq1d ( 𝑥 = 𝐴 → ( ( 𝑀 ‘ ( 𝐹 “ { 𝑥 } ) ) ∈ ( 0 [,) +∞ ) ↔ ( 𝑀 ‘ ( 𝐹 “ { 𝐴 } ) ) ∈ ( 0 [,) +∞ ) ) )
17 16 rspcv ( 𝐴 ∈ ( ran 𝐹 ∖ { 0 } ) → ( ∀ 𝑥 ∈ ( ran 𝐹 ∖ { 0 } ) ( 𝑀 ‘ ( 𝐹 “ { 𝑥 } ) ) ∈ ( 0 [,) +∞ ) → ( 𝑀 ‘ ( 𝐹 “ { 𝐴 } ) ) ∈ ( 0 [,) +∞ ) ) )
18 12 17 mpan9 ( ( 𝜑𝐴 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑀 ‘ ( 𝐹 “ { 𝐴 } ) ) ∈ ( 0 [,) +∞ ) )