Metamath Proof Explorer


Theorem caragenel

Description: Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses caragenel.o ( 𝜑𝑂 ∈ OutMeas )
caragenel.s 𝑆 = ( CaraGen ‘ 𝑂 )
Assertion caragenel ( 𝜑 → ( 𝐸𝑆 ↔ ( 𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀ 𝑎 ∈ 𝒫 dom 𝑂 ( ( 𝑂 ‘ ( 𝑎𝐸 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝐸 ) ) ) = ( 𝑂𝑎 ) ) ) )

Proof

Step Hyp Ref Expression
1 caragenel.o ( 𝜑𝑂 ∈ OutMeas )
2 caragenel.s 𝑆 = ( CaraGen ‘ 𝑂 )
3 caragenval ( 𝑂 ∈ OutMeas → ( CaraGen ‘ 𝑂 ) = { 𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀ 𝑎 ∈ 𝒫 dom 𝑂 ( ( 𝑂 ‘ ( 𝑎𝑒 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝑒 ) ) ) = ( 𝑂𝑎 ) } )
4 1 3 syl ( 𝜑 → ( CaraGen ‘ 𝑂 ) = { 𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀ 𝑎 ∈ 𝒫 dom 𝑂 ( ( 𝑂 ‘ ( 𝑎𝑒 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝑒 ) ) ) = ( 𝑂𝑎 ) } )
5 2 4 syl5eq ( 𝜑𝑆 = { 𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀ 𝑎 ∈ 𝒫 dom 𝑂 ( ( 𝑂 ‘ ( 𝑎𝑒 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝑒 ) ) ) = ( 𝑂𝑎 ) } )
6 5 eleq2d ( 𝜑 → ( 𝐸𝑆𝐸 ∈ { 𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀ 𝑎 ∈ 𝒫 dom 𝑂 ( ( 𝑂 ‘ ( 𝑎𝑒 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝑒 ) ) ) = ( 𝑂𝑎 ) } ) )
7 ineq2 ( 𝑒 = 𝐸 → ( 𝑎𝑒 ) = ( 𝑎𝐸 ) )
8 7 fveq2d ( 𝑒 = 𝐸 → ( 𝑂 ‘ ( 𝑎𝑒 ) ) = ( 𝑂 ‘ ( 𝑎𝐸 ) ) )
9 difeq2 ( 𝑒 = 𝐸 → ( 𝑎𝑒 ) = ( 𝑎𝐸 ) )
10 9 fveq2d ( 𝑒 = 𝐸 → ( 𝑂 ‘ ( 𝑎𝑒 ) ) = ( 𝑂 ‘ ( 𝑎𝐸 ) ) )
11 8 10 oveq12d ( 𝑒 = 𝐸 → ( ( 𝑂 ‘ ( 𝑎𝑒 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝑒 ) ) ) = ( ( 𝑂 ‘ ( 𝑎𝐸 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝐸 ) ) ) )
12 11 eqeq1d ( 𝑒 = 𝐸 → ( ( ( 𝑂 ‘ ( 𝑎𝑒 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝑒 ) ) ) = ( 𝑂𝑎 ) ↔ ( ( 𝑂 ‘ ( 𝑎𝐸 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝐸 ) ) ) = ( 𝑂𝑎 ) ) )
13 12 ralbidv ( 𝑒 = 𝐸 → ( ∀ 𝑎 ∈ 𝒫 dom 𝑂 ( ( 𝑂 ‘ ( 𝑎𝑒 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝑒 ) ) ) = ( 𝑂𝑎 ) ↔ ∀ 𝑎 ∈ 𝒫 dom 𝑂 ( ( 𝑂 ‘ ( 𝑎𝐸 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝐸 ) ) ) = ( 𝑂𝑎 ) ) )
14 13 elrab ( 𝐸 ∈ { 𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀ 𝑎 ∈ 𝒫 dom 𝑂 ( ( 𝑂 ‘ ( 𝑎𝑒 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝑒 ) ) ) = ( 𝑂𝑎 ) } ↔ ( 𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀ 𝑎 ∈ 𝒫 dom 𝑂 ( ( 𝑂 ‘ ( 𝑎𝐸 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝐸 ) ) ) = ( 𝑂𝑎 ) ) )
15 14 a1i ( 𝜑 → ( 𝐸 ∈ { 𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀ 𝑎 ∈ 𝒫 dom 𝑂 ( ( 𝑂 ‘ ( 𝑎𝑒 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝑒 ) ) ) = ( 𝑂𝑎 ) } ↔ ( 𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀ 𝑎 ∈ 𝒫 dom 𝑂 ( ( 𝑂 ‘ ( 𝑎𝐸 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝐸 ) ) ) = ( 𝑂𝑎 ) ) ) )
16 6 15 bitrd ( 𝜑 → ( 𝐸𝑆 ↔ ( 𝐸 ∈ 𝒫 dom 𝑂 ∧ ∀ 𝑎 ∈ 𝒫 dom 𝑂 ( ( 𝑂 ‘ ( 𝑎𝐸 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝐸 ) ) ) = ( 𝑂𝑎 ) ) ) )