Proof of Theorem ipolerval
| Step | Hyp | Ref
| Expression |
|---|
| 1 | | simpl 472 |
. . . . . . 7
⊢ (({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦) → {𝑥, 𝑦} ⊆ 𝐹) |
| 2 | | vex 3176 |
. . . . . . . 8
⊢ 𝑥 ∈ V |
| 3 | | vex 3176 |
. . . . . . . 8
⊢ 𝑦 ∈ V |
| 4 | 2, 3 | prss 4291 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) ↔ {𝑥, 𝑦} ⊆ 𝐹) |
| 5 | 1, 4 | sylibr 223 |
. . . . . 6
⊢ (({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) |
| 6 | 5 | ssopab2i 4928 |
. . . . 5
⊢
{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)} |
| 7 | | df-xp 5044 |
. . . . 5
⊢ (𝐹 × 𝐹) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)} |
| 8 | 6, 7 | sseqtr4i 3601 |
. . . 4
⊢
{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} ⊆ (𝐹 × 𝐹) |
| 9 | | sqxpexg 6861 |
. . . 4
⊢ (𝐹 ∈ 𝑉 → (𝐹 × 𝐹) ∈ V) |
| 10 | | ssexg 4732 |
. . . 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 694 |
. . 3
⊢ (𝐹 ∈ 𝑉 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} ∈ V) |
| 12 | | ipostr 16976 |
. . . 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 15872 |
. . . 4
⊢ le = Slot
(le‘ndx) |
| 14 | | snsspr1 4285 |
. . . . 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 3739 |
. . . . 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 3577 |
. . . 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 15735 |
. . 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 2610 |
. . . 4
⊢
{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} |
| 21 | 19, 20 | ipoval 16977 |
. . 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 6107 |
. 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 2647 |
1
⊢ (𝐹 ∈ 𝑉 → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} = (le‘𝐼)) |
