Step | Hyp | Ref
| Expression |
1 | | dfac8b 10096 |
. . . 4
⊢ (𝐴 ∈ dom card →
∃𝑠 𝑠 We 𝐴) |
2 | 1 | adantr 480 |
. . 3
⊢ ((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) → ∃𝑠 𝑠 We 𝐴) |
3 | | simpll 766 |
. . . . . . 7
⊢ (((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → 𝐴 ∈ dom card) |
4 | | simplr 768 |
. . . . . . . . 9
⊢ (((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) |
5 | | r19.26 3113 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝑆 We 𝐵)) |
6 | 4, 5 | sylib 218 |
. . . . . . . 8
⊢ (((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝑆 We 𝐵)) |
7 | 6 | simpld 494 |
. . . . . . 7
⊢ (((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) |
8 | | iunexg 8000 |
. . . . . . 7
⊢ ((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → ∪
𝑥 ∈ 𝐴 𝐵 ∈ V) |
9 | 3, 7, 8 | syl2anc 583 |
. . . . . 6
⊢ (((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → ∪
𝑥 ∈ 𝐴 𝐵 ∈ V) |
10 | 9, 9 | xpexd 7782 |
. . . . 5
⊢ (((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → (∪ 𝑥 ∈ 𝐴 𝐵 × ∪
𝑥 ∈ 𝐴 𝐵) ∈ V) |
11 | | opabssxp 5791 |
. . . . . 6
⊢
{〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦⦋((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥⦌𝑆𝑧)))} ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 × ∪
𝑥 ∈ 𝐴 𝐵) |
12 | 11 | a1i 11 |
. . . . 5
⊢ (((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦⦋((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥⦌𝑆𝑧)))} ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 × ∪
𝑥 ∈ 𝐴 𝐵)) |
13 | 10, 12 | ssexd 5345 |
. . . 4
⊢ (((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦⦋((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥⦌𝑆𝑧)))} ∈ V) |
14 | | simpr 484 |
. . . . 5
⊢ (((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → 𝑠 We 𝐴) |
15 | | exse 5662 |
. . . . . 6
⊢ (𝐴 ∈ dom card → 𝑠 Se 𝐴) |
16 | 15 | ad2antrr 725 |
. . . . 5
⊢ (((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → 𝑠 Se 𝐴) |
17 | 6 | simprd 495 |
. . . . 5
⊢ (((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → ∀𝑥 ∈ 𝐴 𝑆 We 𝐵) |
18 | | eqid 2734 |
. . . . . 6
⊢ (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢)) = (𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢)) |
19 | | eqid 2734 |
. . . . . 6
⊢
{〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦⦋((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥⦌𝑆𝑧)))} = {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦⦋((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥⦌𝑆𝑧)))} |
20 | 18, 19 | weiunwe 36384 |
. . . . 5
⊢ ((𝑠 We 𝐴 ∧ 𝑠 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 We 𝐵) → {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦⦋((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥⦌𝑆𝑧)))} We ∪
𝑥 ∈ 𝐴 𝐵) |
21 | 14, 16, 17, 20 | syl3anc 1371 |
. . . 4
⊢ (((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦⦋((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥⦌𝑆𝑧)))} We ∪
𝑥 ∈ 𝐴 𝐵) |
22 | | weeq1 5687 |
. . . 4
⊢ (𝑡 = {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦⦋((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥⦌𝑆𝑧)))} → (𝑡 We ∪ 𝑥 ∈ 𝐴 𝐵 ↔ {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦)𝑠((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∨ (((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) = ((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑧) ∧ 𝑦⦋((𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑠𝑢))‘𝑦) / 𝑥⦌𝑆𝑧)))} We ∪
𝑥 ∈ 𝐴 𝐵)) |
23 | 13, 21, 22 | spcedv 3607 |
. . 3
⊢ (((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) ∧ 𝑠 We 𝐴) → ∃𝑡 𝑡 We ∪ 𝑥 ∈ 𝐴 𝐵) |
24 | 2, 23 | exlimddv 1934 |
. 2
⊢ ((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) → ∃𝑡 𝑡 We ∪ 𝑥 ∈ 𝐴 𝐵) |
25 | | ween 10100 |
. 2
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ dom card ↔ ∃𝑡 𝑡 We ∪ 𝑥 ∈ 𝐴 𝐵) |
26 | 24, 25 | sylibr 234 |
1
⊢ ((𝐴 ∈ dom card ∧
∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ dom card) |