Proof of Theorem ipolerval
| Step | Hyp | Ref
| Expression |
| 1 | | simpl 482 |
. . . . . . 7
⊢ (({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦) → {𝑥, 𝑦} ⊆ 𝐹) |
| 2 | | vex 3484 |
. . . . . . . 8
⊢ 𝑥 ∈ V |
| 3 | | vex 3484 |
. . . . . . . 8
⊢ 𝑦 ∈ V |
| 4 | 2, 3 | prss 4820 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) ↔ {𝑥, 𝑦} ⊆ 𝐹) |
| 5 | 1, 4 | sylibr 234 |
. . . . . 6
⊢ (({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) |
| 6 | 5 | ssopab2i 5555 |
. . . . 5
⊢
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)} |
| 7 | | df-xp 5691 |
. . . . 5
⊢ (𝐹 × 𝐹) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)} |
| 8 | 6, 7 | sseqtrri 4033 |
. . . 4
⊢
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} ⊆ (𝐹 × 𝐹) |
| 9 | | sqxpexg 7775 |
. . . 4
⊢ (𝐹 ∈ 𝑉 → (𝐹 × 𝐹) ∈ V) |
| 10 | | ssexg 5323 |
. . . 4
⊢
(({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} ⊆ (𝐹 × 𝐹) ∧ (𝐹 × 𝐹) ∈ V) → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} ∈ V) |
| 11 | 8, 9, 10 | sylancr 587 |
. . 3
⊢ (𝐹 ∈ 𝑉 → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} ∈ V) |
| 12 | | ipostr 18574 |
. . . 4
⊢
({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx),
(ordTop‘{〈𝑥,
𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ∪ {〈(le‘ndx),
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}) Struct 〈1, ;11〉 |
| 13 | | pleid 17411 |
. . . 4
⊢ le = Slot
(le‘ndx) |
| 14 | | snsspr1 4814 |
. . . . 5
⊢
{〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉} ⊆ {〈(le‘ndx),
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉} |
| 15 | | ssun2 4179 |
. . . . 5
⊢
{〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉} ⊆
({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx),
(ordTop‘{〈𝑥,
𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ∪ {〈(le‘ndx),
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}) |
| 16 | 14, 15 | sstri 3993 |
. . . 4
⊢
{〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉} ⊆ ({〈(Base‘ndx),
𝐹〉,
〈(TopSet‘ndx), (ordTop‘{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ∪ {〈(le‘ndx),
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}) |
| 17 | 12, 13, 16 | strfv 17240 |
. . 3
⊢
({〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} ∈ V → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} = (le‘({〈(Base‘ndx),
𝐹〉,
〈(TopSet‘ndx), (ordTop‘{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ∪ {〈(le‘ndx),
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}))) |
| 18 | 11, 17 | syl 17 |
. 2
⊢ (𝐹 ∈ 𝑉 → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} = (le‘({〈(Base‘ndx),
𝐹〉,
〈(TopSet‘ndx), (ordTop‘{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ∪ {〈(le‘ndx),
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}))) |
| 19 | | ipoval.i |
. . . 4
⊢ 𝐼 = (toInc‘𝐹) |
| 20 | | eqid 2737 |
. . . 4
⊢
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} |
| 21 | 19, 20 | ipoval 18575 |
. . 3
⊢ (𝐹 ∈ 𝑉 → 𝐼 = ({〈(Base‘ndx), 𝐹〉,
〈(TopSet‘ndx), (ordTop‘{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ∪ {〈(le‘ndx),
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉})) |
| 22 | 21 | fveq2d 6910 |
. 2
⊢ (𝐹 ∈ 𝑉 → (le‘𝐼) = (le‘({〈(Base‘ndx),
𝐹〉,
〈(TopSet‘ndx), (ordTop‘{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)})〉} ∪ {〈(le‘ndx),
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}))) |
| 23 | 18, 22 | eqtr4d 2780 |
1
⊢ (𝐹 ∈ 𝑉 → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} = (le‘𝐼)) |