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 = ( 0g ‘ 𝑊 )
	sitgval.x	⊢ · = ( ·𝑠 ‘ 𝑊 )
	sitgval.h	⊢ 𝐻 = ( ℝHom ‘ ( Scalar ‘ 𝑊 ) )
	sitgval.1	⊢ ( 𝜑 → 𝑊 ∈ 𝑉 )
	sitgval.2	⊢ ( 𝜑 → 𝑀 ∈ ∪ ran measures )
	sibfmbl.1	⊢ ( 𝜑 → 𝐹 ∈ dom ( 𝑊 sitg 𝑀 ) )
Assertion	sibfima	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 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 = ( 0g ‘ 𝑊 )
5		sitgval.x	⊢ · = ( ·𝑠 ‘ 𝑊 )
6		sitgval.h	⊢ 𝐻 = ( ℝHom ‘ ( Scalar ‘ 𝑊 ) )
7		sitgval.1	⊢ ( 𝜑 → 𝑊 ∈ 𝑉 )
8		sitgval.2	⊢ ( 𝜑 → 𝑀 ∈ ∪ ran measures )
9		sibfmbl.1	⊢ ( 𝜑 → 𝐹 ∈ 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	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( ran 𝐹 ∖ { 0 } ) ) → ( 𝑀 ‘ ( ◡ 𝐹 “ { 𝐴 } ) ) ∈ ( 0 [,) +∞ ) )