Step | Hyp | Ref
| Expression |
1 | | id 22 |
. 2
⊢ (𝑂 ∈ OutMeas → 𝑂 ∈
OutMeas) |
2 | | dmexg 6989 |
. . . . 5
⊢ (𝑂 ∈ OutMeas → dom 𝑂 ∈ V) |
3 | | uniexg 6853 |
. . . . 5
⊢ (dom
𝑂 ∈ V → ∪ dom 𝑂 ∈ V) |
4 | 2, 3 | syl 17 |
. . . 4
⊢ (𝑂 ∈ OutMeas → ∪ dom 𝑂 ∈ V) |
5 | | pwexg 4776 |
. . . 4
⊢ (∪ dom 𝑂 ∈ V → 𝒫 ∪ dom 𝑂 ∈ V) |
6 | 4, 5 | syl 17 |
. . 3
⊢ (𝑂 ∈ OutMeas → 𝒫
∪ dom 𝑂 ∈ V) |
7 | | rabexg 4739 |
. . 3
⊢
(𝒫 ∪ dom 𝑂 ∈ V → {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ∈ V) |
8 | 6, 7 | syl 17 |
. 2
⊢ (𝑂 ∈ OutMeas → {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ∈ V) |
9 | | dmeq 5246 |
. . . . . 6
⊢ (𝑜 = 𝑂 → dom 𝑜 = dom 𝑂) |
10 | 9 | unieqd 4382 |
. . . . 5
⊢ (𝑜 = 𝑂 → ∪ dom
𝑜 = ∪ dom 𝑂) |
11 | 10 | pweqd 4113 |
. . . 4
⊢ (𝑜 = 𝑂 → 𝒫 ∪ dom 𝑜 = 𝒫 ∪ dom
𝑂) |
12 | 11 | raleqdv 3121 |
. . . . 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))) |
13 | | fveq1 6102 |
. . . . . . . 8
⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∩ 𝑒)) = (𝑂‘(𝑎 ∩ 𝑒))) |
14 | | fveq1 6102 |
. . . . . . . 8
⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∖ 𝑒)) = (𝑂‘(𝑎 ∖ 𝑒))) |
15 | 13, 14 | oveq12d 6567 |
. . . . . . 7
⊢ (𝑜 = 𝑂 → ((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒)))) |
16 | | fveq1 6102 |
. . . . . . 7
⊢ (𝑜 = 𝑂 → (𝑜‘𝑎) = (𝑂‘𝑎)) |
17 | 15, 16 | eqeq12d 2625 |
. . . . . 6
⊢ (𝑜 = 𝑂 → (((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎) ↔ ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎))) |
18 | 17 | ralbidv 2969 |
. . . . 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))) |
19 | 12, 18 | bitrd 267 |
. . . 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))) |
20 | 11, 19 | rabeqbidv 3168 |
. . 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)}) |
21 | | df-caragen 39382 |
. . 3
⊢ CaraGen =
(𝑜 ∈ OutMeas ↦
{𝑒 ∈ 𝒫 ∪ dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)}) |
22 | 20, 21 | fvmptg 6189 |
. 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)}) |
23 | 1, 8, 22 | syl2anc 691 |
1
⊢ (𝑂 ∈ OutMeas →
(CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}) |