Metamath Proof Explorer


Theorem elcarsg

Description: Property of being a Caratheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020)

Ref Expression
Hypotheses carsgval.1 ( 𝜑𝑂𝑉 )
carsgval.2 ( 𝜑𝑀 : 𝒫 𝑂 ⟶ ( 0 [,] +∞ ) )
Assertion elcarsg ( 𝜑 → ( 𝐴 ∈ ( toCaraSiga ‘ 𝑀 ) ↔ ( 𝐴𝑂 ∧ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝐴 ) ) ) = ( 𝑀𝑒 ) ) ) )

Proof

Step Hyp Ref Expression
1 carsgval.1 ( 𝜑𝑂𝑉 )
2 carsgval.2 ( 𝜑𝑀 : 𝒫 𝑂 ⟶ ( 0 [,] +∞ ) )
3 1 2 carsgval ( 𝜑 → ( toCaraSiga ‘ 𝑀 ) = { 𝑎 ∈ 𝒫 𝑂 ∣ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝑎 ) ) ) = ( 𝑀𝑒 ) } )
4 3 eleq2d ( 𝜑 → ( 𝐴 ∈ ( toCaraSiga ‘ 𝑀 ) ↔ 𝐴 ∈ { 𝑎 ∈ 𝒫 𝑂 ∣ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝑎 ) ) ) = ( 𝑀𝑒 ) } ) )
5 ineq2 ( 𝑎 = 𝐴 → ( 𝑒𝑎 ) = ( 𝑒𝐴 ) )
6 5 fveq2d ( 𝑎 = 𝐴 → ( 𝑀 ‘ ( 𝑒𝑎 ) ) = ( 𝑀 ‘ ( 𝑒𝐴 ) ) )
7 difeq2 ( 𝑎 = 𝐴 → ( 𝑒𝑎 ) = ( 𝑒𝐴 ) )
8 7 fveq2d ( 𝑎 = 𝐴 → ( 𝑀 ‘ ( 𝑒𝑎 ) ) = ( 𝑀 ‘ ( 𝑒𝐴 ) ) )
9 6 8 oveq12d ( 𝑎 = 𝐴 → ( ( 𝑀 ‘ ( 𝑒𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝑎 ) ) ) = ( ( 𝑀 ‘ ( 𝑒𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝐴 ) ) ) )
10 9 eqeq1d ( 𝑎 = 𝐴 → ( ( ( 𝑀 ‘ ( 𝑒𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝑎 ) ) ) = ( 𝑀𝑒 ) ↔ ( ( 𝑀 ‘ ( 𝑒𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝐴 ) ) ) = ( 𝑀𝑒 ) ) )
11 10 ralbidv ( 𝑎 = 𝐴 → ( ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝑎 ) ) ) = ( 𝑀𝑒 ) ↔ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝐴 ) ) ) = ( 𝑀𝑒 ) ) )
12 11 elrab ( 𝐴 ∈ { 𝑎 ∈ 𝒫 𝑂 ∣ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝑎 ) ) ) = ( 𝑀𝑒 ) } ↔ ( 𝐴 ∈ 𝒫 𝑂 ∧ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝐴 ) ) ) = ( 𝑀𝑒 ) ) )
13 elex ( 𝐴 ∈ 𝒫 𝑂𝐴 ∈ V )
14 13 a1i ( 𝜑 → ( 𝐴 ∈ 𝒫 𝑂𝐴 ∈ V ) )
15 1 adantr ( ( 𝜑𝐴𝑂 ) → 𝑂𝑉 )
16 simpr ( ( 𝜑𝐴𝑂 ) → 𝐴𝑂 )
17 15 16 ssexd ( ( 𝜑𝐴𝑂 ) → 𝐴 ∈ V )
18 17 ex ( 𝜑 → ( 𝐴𝑂𝐴 ∈ V ) )
19 elpwg ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝑂𝐴𝑂 ) )
20 19 a1i ( 𝜑 → ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝑂𝐴𝑂 ) ) )
21 14 18 20 pm5.21ndd ( 𝜑 → ( 𝐴 ∈ 𝒫 𝑂𝐴𝑂 ) )
22 21 anbi1d ( 𝜑 → ( ( 𝐴 ∈ 𝒫 𝑂 ∧ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝐴 ) ) ) = ( 𝑀𝑒 ) ) ↔ ( 𝐴𝑂 ∧ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝐴 ) ) ) = ( 𝑀𝑒 ) ) ) )
23 12 22 syl5bb ( 𝜑 → ( 𝐴 ∈ { 𝑎 ∈ 𝒫 𝑂 ∣ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝑎 ) ) ) = ( 𝑀𝑒 ) } ↔ ( 𝐴𝑂 ∧ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝐴 ) ) ) = ( 𝑀𝑒 ) ) ) )
24 4 23 bitrd ( 𝜑 → ( 𝐴 ∈ ( toCaraSiga ‘ 𝑀 ) ↔ ( 𝐴𝑂 ∧ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒𝐴 ) ) ) = ( 𝑀𝑒 ) ) ) )