Step |
Hyp |
Ref |
Expression |
1 |
|
caragenuncl.1 |
⊢ ( 𝜑 → 𝑂 ∈ OutMeas ) |
2 |
|
caragenuncl.2 |
⊢ 𝑆 = ( CaraGen ‘ 𝑂 ) |
3 |
|
caragenuncl.3 |
⊢ ( 𝜑 → 𝐸 ∈ 𝑆 ) |
4 |
|
caragenuncl.4 |
⊢ ( 𝜑 → 𝐹 ∈ 𝑆 ) |
5 |
|
eqid |
⊢ ∪ dom 𝑂 = ∪ dom 𝑂 |
6 |
1 2 3 5
|
caragenelss |
⊢ ( 𝜑 → 𝐸 ⊆ ∪ dom 𝑂 ) |
7 |
1 2 4 5
|
caragenelss |
⊢ ( 𝜑 → 𝐹 ⊆ ∪ dom 𝑂 ) |
8 |
6 7
|
unssd |
⊢ ( 𝜑 → ( 𝐸 ∪ 𝐹 ) ⊆ ∪ dom 𝑂 ) |
9 |
1 5
|
unidmex |
⊢ ( 𝜑 → ∪ dom 𝑂 ∈ V ) |
10 |
|
ssexg |
⊢ ( ( ( 𝐸 ∪ 𝐹 ) ⊆ ∪ dom 𝑂 ∧ ∪ dom 𝑂 ∈ V ) → ( 𝐸 ∪ 𝐹 ) ∈ V ) |
11 |
8 9 10
|
syl2anc |
⊢ ( 𝜑 → ( 𝐸 ∪ 𝐹 ) ∈ V ) |
12 |
|
elpwg |
⊢ ( ( 𝐸 ∪ 𝐹 ) ∈ V → ( ( 𝐸 ∪ 𝐹 ) ∈ 𝒫 ∪ dom 𝑂 ↔ ( 𝐸 ∪ 𝐹 ) ⊆ ∪ dom 𝑂 ) ) |
13 |
11 12
|
syl |
⊢ ( 𝜑 → ( ( 𝐸 ∪ 𝐹 ) ∈ 𝒫 ∪ dom 𝑂 ↔ ( 𝐸 ∪ 𝐹 ) ⊆ ∪ dom 𝑂 ) ) |
14 |
8 13
|
mpbird |
⊢ ( 𝜑 → ( 𝐸 ∪ 𝐹 ) ∈ 𝒫 ∪ dom 𝑂 ) |
15 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → 𝑂 ∈ OutMeas ) |
16 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → 𝐸 ∈ 𝑆 ) |
17 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → 𝐹 ∈ 𝑆 ) |
18 |
|
elpwi |
⊢ ( 𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂 ) |
19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → 𝑎 ⊆ ∪ dom 𝑂 ) |
20 |
15 2 16 17 5 19
|
caragenuncllem |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ) → ( ( 𝑂 ‘ ( 𝑎 ∩ ( 𝐸 ∪ 𝐹 ) ) ) +𝑒 ( 𝑂 ‘ ( 𝑎 ∖ ( 𝐸 ∪ 𝐹 ) ) ) ) = ( 𝑂 ‘ 𝑎 ) ) |
21 |
1 5 2 14 20
|
carageneld |
⊢ ( 𝜑 → ( 𝐸 ∪ 𝐹 ) ∈ 𝑆 ) |