| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | id 22 | . 2
⊢ (𝑂 ∈ OutMeas → 𝑂 ∈
OutMeas) | 
| 2 |  | dmexg 7923 | . . . . 5
⊢ (𝑂 ∈ OutMeas → dom 𝑂 ∈ V) | 
| 3 | 2 | uniexd 7762 | . . . 4
⊢ (𝑂 ∈ OutMeas → ∪ dom 𝑂 ∈ V) | 
| 4 | 3 | pwexd 5379 | . . 3
⊢ (𝑂 ∈ OutMeas → 𝒫
∪ dom 𝑂 ∈ V) | 
| 5 |  | rabexg 5337 | . . 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 5914 | . . . . . 6
⊢ (𝑜 = 𝑂 → dom 𝑜 = dom 𝑂) | 
| 8 | 7 | unieqd 4920 | . . . . 5
⊢ (𝑜 = 𝑂 → ∪ dom
𝑜 = ∪ dom 𝑂) | 
| 9 | 8 | pweqd 4617 | . . . 4
⊢ (𝑜 = 𝑂 → 𝒫 ∪ dom 𝑜 = 𝒫 ∪ dom
𝑂) | 
| 10 | 9 | raleqdv 3326 | . . . . 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 6905 | . . . . . . . 8
⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∩ 𝑒)) = (𝑂‘(𝑎 ∩ 𝑒))) | 
| 12 |  | fveq1 6905 | . . . . . . . 8
⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∖ 𝑒)) = (𝑂‘(𝑎 ∖ 𝑒))) | 
| 13 | 11, 12 | oveq12d 7449 | . . . . . . 7
⊢ (𝑜 = 𝑂 → ((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒)))) | 
| 14 |  | fveq1 6905 | . . . . . . 7
⊢ (𝑜 = 𝑂 → (𝑜‘𝑎) = (𝑂‘𝑎)) | 
| 15 | 13, 14 | eqeq12d 2753 | . . . . . 6
⊢ (𝑜 = 𝑂 → (((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎) ↔ ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎))) | 
| 16 | 15 | ralbidv 3178 | . . . . 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 279 | . . . 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 3455 | . . 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 46507 | . . 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 7014 | . 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 584 | 1
⊢ (𝑂 ∈ OutMeas →
(CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)}) |