Metamath Proof Explorer


Theorem caragenuncl

Description: The Caratheodory's construction is closed under the union. Step (c) in the proof of Theorem 113C of Fremlin1 p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses caragenuncl.1 ( 𝜑𝑂 ∈ OutMeas )
caragenuncl.2 𝑆 = ( CaraGen ‘ 𝑂 )
caragenuncl.3 ( 𝜑𝐸𝑆 )
caragenuncl.4 ( 𝜑𝐹𝑆 )
Assertion caragenuncl ( 𝜑 → ( 𝐸𝐹 ) ∈ 𝑆 )

Proof

Step Hyp Ref Expression
1 caragenuncl.1 ( 𝜑𝑂 ∈ OutMeas )
2 caragenuncl.2 𝑆 = ( CaraGen ‘ 𝑂 )
3 caragenuncl.3 ( 𝜑𝐸𝑆 )
4 caragenuncl.4 ( 𝜑𝐹𝑆 )
5 eqid dom 𝑂 = dom 𝑂
6 1 2 3 5 caragenelss ( 𝜑𝐸 dom 𝑂 )
7 1 2 4 5 caragenelss ( 𝜑𝐹 dom 𝑂 )
8 6 7 unssd ( 𝜑 → ( 𝐸𝐹 ) ⊆ dom 𝑂 )
9 1 5 unidmex ( 𝜑 dom 𝑂 ∈ V )
10 ssexg ( ( ( 𝐸𝐹 ) ⊆ dom 𝑂 dom 𝑂 ∈ V ) → ( 𝐸𝐹 ) ∈ V )
11 8 9 10 syl2anc ( 𝜑 → ( 𝐸𝐹 ) ∈ V )
12 elpwg ( ( 𝐸𝐹 ) ∈ V → ( ( 𝐸𝐹 ) ∈ 𝒫 dom 𝑂 ↔ ( 𝐸𝐹 ) ⊆ dom 𝑂 ) )
13 11 12 syl ( 𝜑 → ( ( 𝐸𝐹 ) ∈ 𝒫 dom 𝑂 ↔ ( 𝐸𝐹 ) ⊆ dom 𝑂 ) )
14 8 13 mpbird ( 𝜑 → ( 𝐸𝐹 ) ∈ 𝒫 dom 𝑂 )
15 1 adantr ( ( 𝜑𝑎 ∈ 𝒫 dom 𝑂 ) → 𝑂 ∈ OutMeas )
16 3 adantr ( ( 𝜑𝑎 ∈ 𝒫 dom 𝑂 ) → 𝐸𝑆 )
17 4 adantr ( ( 𝜑𝑎 ∈ 𝒫 dom 𝑂 ) → 𝐹𝑆 )
18 elpwi ( 𝑎 ∈ 𝒫 dom 𝑂𝑎 dom 𝑂 )
19 18 adantl ( ( 𝜑𝑎 ∈ 𝒫 dom 𝑂 ) → 𝑎 dom 𝑂 )
20 15 2 16 17 5 19 caragenuncllem ( ( 𝜑𝑎 ∈ 𝒫 dom 𝑂 ) → ( ( 𝑂 ‘ ( 𝑎 ∩ ( 𝐸𝐹 ) ) ) +𝑒 ( 𝑂 ‘ ( 𝑎 ∖ ( 𝐸𝐹 ) ) ) ) = ( 𝑂𝑎 ) )
21 1 5 2 14 20 carageneld ( 𝜑 → ( 𝐸𝐹 ) ∈ 𝑆 )