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