Step | Hyp | Ref
| Expression |
1 | | id 22 |
. 2
⊢ (𝑂 ∈ OutMeas → 𝑂 ∈
OutMeas) |
2 | | dmexg 7690 |
. . . . 5
⊢ (𝑂 ∈ OutMeas → dom 𝑂 ∈ V) |
3 | 2 | uniexd 7539 |
. . . 4
⊢ (𝑂 ∈ OutMeas → ∪ dom 𝑂 ∈ V) |
4 | 3 | pwexd 5281 |
. . 3
⊢ (𝑂 ∈ OutMeas → 𝒫
∪ dom 𝑂 ∈ V) |
5 | | rabexg 5233 |
. . 3
⊢
(𝒫 ∪ dom 𝑂 ∈ V → {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ∈ V) |
6 | 4, 5 | syl 17 |
. 2
⊢ (𝑂 ∈ OutMeas → {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ∈ V) |
7 | | dmeq 5781 |
. . . . . 6
⊢ (𝑜 = 𝑂 → dom 𝑜 = dom 𝑂) |
8 | 7 | unieqd 4842 |
. . . . 5
⊢ (𝑜 = 𝑂 → ∪ dom
𝑜 = ∪ dom 𝑂) |
9 | 8 | pweqd 4541 |
. . . 4
⊢ (𝑜 = 𝑂 → 𝒫 ∪ dom 𝑜 = 𝒫 ∪ dom
𝑂) |
10 | 9 | raleqdv 3332 |
. . . . 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))) |
11 | | fveq1 6725 |
. . . . . . . 8
⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∩ 𝑒)) = (𝑂‘(𝑎 ∩ 𝑒))) |
12 | | fveq1 6725 |
. . . . . . . 8
⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∖ 𝑒)) = (𝑂‘(𝑎 ∖ 𝑒))) |
13 | 11, 12 | oveq12d 7240 |
. . . . . . 7
⊢ (𝑜 = 𝑂 → ((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒)))) |
14 | | fveq1 6725 |
. . . . . . 7
⊢ (𝑜 = 𝑂 → (𝑜‘𝑎) = (𝑂‘𝑎)) |
15 | 13, 14 | eqeq12d 2754 |
. . . . . 6
⊢ (𝑜 = 𝑂 → (((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎) ↔ ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎))) |
16 | 15 | ralbidv 3119 |
. . . . 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))) |
17 | 10, 16 | bitrd 282 |
. . . 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))) |
18 | 9, 17 | rabeqbidv 3403 |
. . 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)}) |
19 | | df-caragen 43720 |
. . 3
⊢ CaraGen =
(𝑜 ∈ OutMeas ↦
{𝑒 ∈ 𝒫 ∪ dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)}) |
20 | 18, 19 | fvmptg 6825 |
. 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)}) |
21 | 1, 6, 20 | syl2anc 587 |
1
⊢ (𝑂 ∈ OutMeas →
(CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}) |