Metamath Proof Explorer


Theorem caragenunidm

Description: The base set of an outer measure belongs to the sigma-algebra generated by the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses caragenunidm.o ( 𝜑𝑂 ∈ OutMeas )
caragenunidm.x 𝑋 = dom 𝑂
caragenunidm.s 𝑆 = ( CaraGen ‘ 𝑂 )
Assertion caragenunidm ( 𝜑𝑋𝑆 )

Proof

Step Hyp Ref Expression
1 caragenunidm.o ( 𝜑𝑂 ∈ OutMeas )
2 caragenunidm.x 𝑋 = dom 𝑂
3 caragenunidm.s 𝑆 = ( CaraGen ‘ 𝑂 )
4 dmexg ( 𝑂 ∈ OutMeas → dom 𝑂 ∈ V )
5 uniexg ( dom 𝑂 ∈ V → dom 𝑂 ∈ V )
6 1 4 5 3syl ( 𝜑 dom 𝑂 ∈ V )
7 2 6 eqeltrid ( 𝜑𝑋 ∈ V )
8 pwidg ( 𝑋 ∈ V → 𝑋 ∈ 𝒫 𝑋 )
9 7 8 syl ( 𝜑𝑋 ∈ 𝒫 𝑋 )
10 elpwi ( 𝑎 ∈ 𝒫 𝑋𝑎𝑋 )
11 df-ss ( 𝑎𝑋 ↔ ( 𝑎𝑋 ) = 𝑎 )
12 11 biimpi ( 𝑎𝑋 → ( 𝑎𝑋 ) = 𝑎 )
13 10 12 syl ( 𝑎 ∈ 𝒫 𝑋 → ( 𝑎𝑋 ) = 𝑎 )
14 13 fveq2d ( 𝑎 ∈ 𝒫 𝑋 → ( 𝑂 ‘ ( 𝑎𝑋 ) ) = ( 𝑂𝑎 ) )
15 14 adantl ( ( 𝜑𝑎 ∈ 𝒫 𝑋 ) → ( 𝑂 ‘ ( 𝑎𝑋 ) ) = ( 𝑂𝑎 ) )
16 ssdif0 ( 𝑎𝑋 ↔ ( 𝑎𝑋 ) = ∅ )
17 10 16 sylib ( 𝑎 ∈ 𝒫 𝑋 → ( 𝑎𝑋 ) = ∅ )
18 17 fveq2d ( 𝑎 ∈ 𝒫 𝑋 → ( 𝑂 ‘ ( 𝑎𝑋 ) ) = ( 𝑂 ‘ ∅ ) )
19 18 adantl ( ( 𝜑𝑎 ∈ 𝒫 𝑋 ) → ( 𝑂 ‘ ( 𝑎𝑋 ) ) = ( 𝑂 ‘ ∅ ) )
20 1 ome0 ( 𝜑 → ( 𝑂 ‘ ∅ ) = 0 )
21 20 adantr ( ( 𝜑𝑎 ∈ 𝒫 𝑋 ) → ( 𝑂 ‘ ∅ ) = 0 )
22 19 21 eqtrd ( ( 𝜑𝑎 ∈ 𝒫 𝑋 ) → ( 𝑂 ‘ ( 𝑎𝑋 ) ) = 0 )
23 15 22 oveq12d ( ( 𝜑𝑎 ∈ 𝒫 𝑋 ) → ( ( 𝑂 ‘ ( 𝑎𝑋 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝑋 ) ) ) = ( ( 𝑂𝑎 ) +𝑒 0 ) )
24 iccssxr ( 0 [,] +∞ ) ⊆ ℝ*
25 1 adantr ( ( 𝜑𝑎 ∈ 𝒫 𝑋 ) → 𝑂 ∈ OutMeas )
26 10 adantl ( ( 𝜑𝑎 ∈ 𝒫 𝑋 ) → 𝑎𝑋 )
27 25 2 26 omecl ( ( 𝜑𝑎 ∈ 𝒫 𝑋 ) → ( 𝑂𝑎 ) ∈ ( 0 [,] +∞ ) )
28 24 27 sselid ( ( 𝜑𝑎 ∈ 𝒫 𝑋 ) → ( 𝑂𝑎 ) ∈ ℝ* )
29 28 xaddid1d ( ( 𝜑𝑎 ∈ 𝒫 𝑋 ) → ( ( 𝑂𝑎 ) +𝑒 0 ) = ( 𝑂𝑎 ) )
30 eqidd ( ( 𝜑𝑎 ∈ 𝒫 𝑋 ) → ( 𝑂𝑎 ) = ( 𝑂𝑎 ) )
31 23 29 30 3eqtrd ( ( 𝜑𝑎 ∈ 𝒫 𝑋 ) → ( ( 𝑂 ‘ ( 𝑎𝑋 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎𝑋 ) ) ) = ( 𝑂𝑎 ) )
32 1 2 3 9 31 carageneld ( 𝜑𝑋𝑆 )