Step | Hyp | Ref
| Expression |
1 | | id 22 |
. 2
⊢ (𝑂 ∈ OutMeas → 𝑂 ∈
OutMeas) |
2 | | dmexg 7358 |
. . . . 5
⊢ (𝑂 ∈ OutMeas → dom 𝑂 ∈ V) |
3 | | uniexg 7215 |
. . . . 5
⊢ (dom
𝑂 ∈ V → ∪ dom 𝑂 ∈ V) |
4 | 2, 3 | syl 17 |
. . . 4
⊢ (𝑂 ∈ OutMeas → ∪ dom 𝑂 ∈ V) |
5 | 4 | pwexd 5079 |
. . 3
⊢ (𝑂 ∈ OutMeas → 𝒫
∪ dom 𝑂 ∈ V) |
6 | | rabexg 5036 |
. . 3
⊢
(𝒫 ∪ dom 𝑂 ∈ V → {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ∈ V) |
7 | 5, 6 | syl 17 |
. 2
⊢ (𝑂 ∈ OutMeas → {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ∈ V) |
8 | | dmeq 5556 |
. . . . . 6
⊢ (𝑜 = 𝑂 → dom 𝑜 = dom 𝑂) |
9 | 8 | unieqd 4668 |
. . . . 5
⊢ (𝑜 = 𝑂 → ∪ dom
𝑜 = ∪ dom 𝑂) |
10 | 9 | pweqd 4383 |
. . . 4
⊢ (𝑜 = 𝑂 → 𝒫 ∪ dom 𝑜 = 𝒫 ∪ dom
𝑂) |
11 | 10 | raleqdv 3356 |
. . . . 5
⊢ (𝑜 = 𝑂 → (∀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))) |
12 | | fveq1 6432 |
. . . . . . . 8
⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∩ 𝑒)) = (𝑂‘(𝑎 ∩ 𝑒))) |
13 | | fveq1 6432 |
. . . . . . . 8
⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∖ 𝑒)) = (𝑂‘(𝑎 ∖ 𝑒))) |
14 | 12, 13 | oveq12d 6923 |
. . . . . . 7
⊢ (𝑜 = 𝑂 → ((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒)))) |
15 | | fveq1 6432 |
. . . . . . 7
⊢ (𝑜 = 𝑂 → (𝑜‘𝑎) = (𝑂‘𝑎)) |
16 | 14, 15 | eqeq12d 2840 |
. . . . . 6
⊢ (𝑜 = 𝑂 → (((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎) ↔ ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎))) |
17 | 16 | ralbidv 3195 |
. . . . 5
⊢ (𝑜 = 𝑂 → (∀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))) |
18 | 11, 17 | bitrd 271 |
. . . 4
⊢ (𝑜 = 𝑂 → (∀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))) |
19 | 10, 18 | rabeqbidv 3408 |
. . 3
⊢ (𝑜 = 𝑂 → {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)}) |
20 | | df-caragen 41493 |
. . 3
⊢ CaraGen =
(𝑜 ∈ OutMeas ↦
{𝑒 ∈ 𝒫 ∪ dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)}) |
21 | 19, 20 | fvmptg 6527 |
. 2
⊢ ((𝑂 ∈ OutMeas ∧ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ∈ V) → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}) |
22 | 1, 7, 21 | syl2anc 579 |
1
⊢ (𝑂 ∈ OutMeas →
(CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}) |