| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpl 482 | . . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → 𝑟 = 𝑅) | 
| 2 | 1 | dmeqd 5916 | . . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → dom 𝑟 = dom 𝑅) | 
| 3 |  | simpr 484 | . . . . . . . . 9
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → 𝑚 = 𝑀) | 
| 4 | 3 | dmeqd 5916 | . . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → dom 𝑚 = dom 𝑀) | 
| 5 | 4 | unieqd 4920 | . . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ∪ dom
𝑚 = ∪ dom 𝑀) | 
| 6 | 2, 5 | oveq12d 7449 | . . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (dom 𝑟 ↑m ∪ dom 𝑚) = (dom 𝑅 ↑m ∪ dom 𝑀)) | 
| 7 | 6 | eleq2d 2827 | . . . . 5
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (𝑓 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ↔ 𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀))) | 
| 8 | 6 | eleq2d 2827 | . . . . 5
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (𝑔 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ↔ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀))) | 
| 9 | 7, 8 | anbi12d 632 | . . . 4
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ((𝑓 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟 ↑m ∪ dom 𝑚)) ↔ (𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)))) | 
| 10 | 1 | breqd 5154 | . . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ((𝑓‘𝑥)𝑟(𝑔‘𝑥) ↔ (𝑓‘𝑥)𝑅(𝑔‘𝑥))) | 
| 11 | 5, 10 | rabeqbidv 3455 | . . . . 5
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → {𝑥 ∈ ∪ dom
𝑚 ∣ (𝑓‘𝑥)𝑟(𝑔‘𝑥)} = {𝑥 ∈ ∪ dom
𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}) | 
| 12 | 11, 3 | breq12d 5156 | . . . 4
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → ({𝑥 ∈ ∪ dom
𝑚 ∣ (𝑓‘𝑥)𝑟(𝑔‘𝑥)}a.e.𝑚 ↔ {𝑥 ∈ ∪ dom
𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)) | 
| 13 | 9, 12 | anbi12d 632 | . . 3
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → (((𝑓 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟 ↑m ∪ dom 𝑚)) ∧ {𝑥 ∈ ∪ dom
𝑚 ∣ (𝑓‘𝑥)𝑟(𝑔‘𝑥)}a.e.𝑚) ↔ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom
𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀))) | 
| 14 | 13 | opabbidv 5209 | . 2
⊢ ((𝑟 = 𝑅 ∧ 𝑚 = 𝑀) → {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟 ↑m ∪ dom 𝑚)) ∧ {𝑥 ∈ ∪ dom
𝑚 ∣ (𝑓‘𝑥)𝑟(𝑔‘𝑥)}a.e.𝑚)} = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom
𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)}) | 
| 15 |  | df-fae 34246 | . 2
⊢ ~ a.e. =
(𝑟 ∈ V, 𝑚 ∈ ∪ ran measures ↦ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (dom 𝑟 ↑m ∪ dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟 ↑m ∪ dom 𝑚)) ∧ {𝑥 ∈ ∪ dom
𝑚 ∣ (𝑓‘𝑥)𝑟(𝑔‘𝑥)}a.e.𝑚)}) | 
| 16 |  | ovex 7464 | . . . 4
⊢ (dom
𝑅 ↑m ∪ dom 𝑀) ∈ V | 
| 17 | 16, 16 | xpex 7773 | . . 3
⊢ ((dom
𝑅 ↑m ∪ dom 𝑀) × (dom 𝑅 ↑m ∪ dom 𝑀)) ∈ V | 
| 18 |  | opabssxp 5778 | . . 3
⊢
{〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom
𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)} ⊆ ((dom 𝑅 ↑m ∪ dom 𝑀) × (dom 𝑅 ↑m ∪ dom 𝑀)) | 
| 19 | 17, 18 | ssexi 5322 | . 2
⊢
{〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom
𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)} ∈ V | 
| 20 | 14, 15, 19 | ovmpoa 7588 | 1
⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ ∪ ran measures) → (𝑅~ a.e.𝑀) = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (dom 𝑅 ↑m ∪ dom 𝑀) ∧ 𝑔 ∈ (dom 𝑅 ↑m ∪ dom 𝑀)) ∧ {𝑥 ∈ ∪ dom
𝑀 ∣ (𝑓‘𝑥)𝑅(𝑔‘𝑥)}a.e.𝑀)}) |