Step |
Hyp |
Ref |
Expression |
1 |
|
carsgval.1 |
⊢ ( 𝜑 → 𝑂 ∈ 𝑉 ) |
2 |
|
carsgval.2 |
⊢ ( 𝜑 → 𝑀 : 𝒫 𝑂 ⟶ ( 0 [,] +∞ ) ) |
3 |
1 2
|
carsgval |
⊢ ( 𝜑 → ( toCaraSiga ‘ 𝑀 ) = { 𝑎 ∈ 𝒫 𝑂 ∣ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒 ∩ 𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝑎 ) ) ) = ( 𝑀 ‘ 𝑒 ) } ) |
4 |
3
|
eleq2d |
⊢ ( 𝜑 → ( 𝐴 ∈ ( toCaraSiga ‘ 𝑀 ) ↔ 𝐴 ∈ { 𝑎 ∈ 𝒫 𝑂 ∣ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒 ∩ 𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝑎 ) ) ) = ( 𝑀 ‘ 𝑒 ) } ) ) |
5 |
|
ineq2 |
⊢ ( 𝑎 = 𝐴 → ( 𝑒 ∩ 𝑎 ) = ( 𝑒 ∩ 𝐴 ) ) |
6 |
5
|
fveq2d |
⊢ ( 𝑎 = 𝐴 → ( 𝑀 ‘ ( 𝑒 ∩ 𝑎 ) ) = ( 𝑀 ‘ ( 𝑒 ∩ 𝐴 ) ) ) |
7 |
|
difeq2 |
⊢ ( 𝑎 = 𝐴 → ( 𝑒 ∖ 𝑎 ) = ( 𝑒 ∖ 𝐴 ) ) |
8 |
7
|
fveq2d |
⊢ ( 𝑎 = 𝐴 → ( 𝑀 ‘ ( 𝑒 ∖ 𝑎 ) ) = ( 𝑀 ‘ ( 𝑒 ∖ 𝐴 ) ) ) |
9 |
6 8
|
oveq12d |
⊢ ( 𝑎 = 𝐴 → ( ( 𝑀 ‘ ( 𝑒 ∩ 𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝑎 ) ) ) = ( ( 𝑀 ‘ ( 𝑒 ∩ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝐴 ) ) ) ) |
10 |
9
|
eqeq1d |
⊢ ( 𝑎 = 𝐴 → ( ( ( 𝑀 ‘ ( 𝑒 ∩ 𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝑎 ) ) ) = ( 𝑀 ‘ 𝑒 ) ↔ ( ( 𝑀 ‘ ( 𝑒 ∩ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝐴 ) ) ) = ( 𝑀 ‘ 𝑒 ) ) ) |
11 |
10
|
ralbidv |
⊢ ( 𝑎 = 𝐴 → ( ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒 ∩ 𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝑎 ) ) ) = ( 𝑀 ‘ 𝑒 ) ↔ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒 ∩ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝐴 ) ) ) = ( 𝑀 ‘ 𝑒 ) ) ) |
12 |
11
|
elrab |
⊢ ( 𝐴 ∈ { 𝑎 ∈ 𝒫 𝑂 ∣ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒 ∩ 𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝑎 ) ) ) = ( 𝑀 ‘ 𝑒 ) } ↔ ( 𝐴 ∈ 𝒫 𝑂 ∧ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒 ∩ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝐴 ) ) ) = ( 𝑀 ‘ 𝑒 ) ) ) |
13 |
|
elex |
⊢ ( 𝐴 ∈ 𝒫 𝑂 → 𝐴 ∈ V ) |
14 |
13
|
a1i |
⊢ ( 𝜑 → ( 𝐴 ∈ 𝒫 𝑂 → 𝐴 ∈ V ) ) |
15 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝑂 ) → 𝑂 ∈ 𝑉 ) |
16 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝑂 ) → 𝐴 ⊆ 𝑂 ) |
17 |
15 16
|
ssexd |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ 𝑂 ) → 𝐴 ∈ V ) |
18 |
17
|
ex |
⊢ ( 𝜑 → ( 𝐴 ⊆ 𝑂 → 𝐴 ∈ V ) ) |
19 |
|
elpwg |
⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝑂 ↔ 𝐴 ⊆ 𝑂 ) ) |
20 |
19
|
a1i |
⊢ ( 𝜑 → ( 𝐴 ∈ V → ( 𝐴 ∈ 𝒫 𝑂 ↔ 𝐴 ⊆ 𝑂 ) ) ) |
21 |
14 18 20
|
pm5.21ndd |
⊢ ( 𝜑 → ( 𝐴 ∈ 𝒫 𝑂 ↔ 𝐴 ⊆ 𝑂 ) ) |
22 |
21
|
anbi1d |
⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝒫 𝑂 ∧ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒 ∩ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝐴 ) ) ) = ( 𝑀 ‘ 𝑒 ) ) ↔ ( 𝐴 ⊆ 𝑂 ∧ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒 ∩ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝐴 ) ) ) = ( 𝑀 ‘ 𝑒 ) ) ) ) |
23 |
12 22
|
syl5bb |
⊢ ( 𝜑 → ( 𝐴 ∈ { 𝑎 ∈ 𝒫 𝑂 ∣ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒 ∩ 𝑎 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝑎 ) ) ) = ( 𝑀 ‘ 𝑒 ) } ↔ ( 𝐴 ⊆ 𝑂 ∧ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒 ∩ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝐴 ) ) ) = ( 𝑀 ‘ 𝑒 ) ) ) ) |
24 |
4 23
|
bitrd |
⊢ ( 𝜑 → ( 𝐴 ∈ ( toCaraSiga ‘ 𝑀 ) ↔ ( 𝐴 ⊆ 𝑂 ∧ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒 ∩ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ 𝐴 ) ) ) = ( 𝑀 ‘ 𝑒 ) ) ) ) |