Step | Hyp | Ref
| Expression |
1 | | elex 3447 |
. 2
⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) |
2 | | ipoval.i |
. . 3
⊢ 𝐼 = (toInc‘𝐹) |
3 | | vex 3433 |
. . . . . . . 8
⊢ 𝑓 ∈ V |
4 | 3, 3 | xpex 7593 |
. . . . . . 7
⊢ (𝑓 × 𝑓) ∈ V |
5 | | simpl 483 |
. . . . . . . . . 10
⊢ (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) → {𝑥, 𝑦} ⊆ 𝑓) |
6 | | vex 3433 |
. . . . . . . . . . 11
⊢ 𝑥 ∈ V |
7 | | vex 3433 |
. . . . . . . . . . 11
⊢ 𝑦 ∈ V |
8 | 6, 7 | prss 4753 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓) ↔ {𝑥, 𝑦} ⊆ 𝑓) |
9 | 5, 8 | sylibr 233 |
. . . . . . . . 9
⊢ (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)) |
10 | 9 | ssopab2i 5460 |
. . . . . . . 8
⊢
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)} |
11 | | df-xp 5590 |
. . . . . . . 8
⊢ (𝑓 × 𝑓) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)} |
12 | 10, 11 | sseqtrri 3957 |
. . . . . . 7
⊢
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ⊆ (𝑓 × 𝑓) |
13 | 4, 12 | ssexi 5244 |
. . . . . 6
⊢
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ∈ V |
14 | 13 | a1i 11 |
. . . . 5
⊢ (𝑓 = 𝐹 → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ∈ V) |
15 | | sseq2 3946 |
. . . . . . . 8
⊢ (𝑓 = 𝐹 → ({𝑥, 𝑦} ⊆ 𝑓 ↔ {𝑥, 𝑦} ⊆ 𝐹)) |
16 | 15 | anbi1d 630 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦))) |
17 | 16 | opabbidv 5139 |
. . . . . 6
⊢ (𝑓 = 𝐹 → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)}) |
18 | | ipoval.l |
. . . . . 6
⊢ ≤ =
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦)} |
19 | 17, 18 | eqtr4di 2796 |
. . . . 5
⊢ (𝑓 = 𝐹 → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} = ≤ ) |
20 | | simpl 483 |
. . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → 𝑓 = 𝐹) |
21 | 20 | opeq2d 4811 |
. . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
〈(Base‘ndx), 𝑓〉 = 〈(Base‘ndx), 𝐹〉) |
22 | | simpr 485 |
. . . . . . . . 9
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → 𝑜 = ≤ ) |
23 | 22 | fveq2d 6770 |
. . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
(ordTop‘𝑜) =
(ordTop‘ ≤ )) |
24 | 23 | opeq2d 4811 |
. . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
〈(TopSet‘ndx), (ordTop‘𝑜)〉 = 〈(TopSet‘ndx),
(ordTop‘ ≤
)〉) |
25 | 21, 24 | preq12d 4677 |
. . . . . 6
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
{〈(Base‘ndx), 𝑓〉, 〈(TopSet‘ndx),
(ordTop‘𝑜)〉} =
{〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx),
(ordTop‘ ≤
)〉}) |
26 | 22 | opeq2d 4811 |
. . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
〈(le‘ndx), 𝑜〉 = 〈(le‘ndx), ≤
〉) |
27 | | id 22 |
. . . . . . . . . 10
⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹) |
28 | | rabeq 3415 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝐹 → {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅} = {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅}) |
29 | 28 | unieqd 4853 |
. . . . . . . . . 10
⊢ (𝑓 = 𝐹 → ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅} = ∪
{𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅}) |
30 | 27, 29 | mpteq12dv 5164 |
. . . . . . . . 9
⊢ (𝑓 = 𝐹 → (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅}) = (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})) |
31 | 30 | adantr 481 |
. . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅}) = (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})) |
32 | 31 | opeq2d 4811 |
. . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
〈(oc‘ndx), (𝑥
∈ 𝑓 ↦ ∪ {𝑦
∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})〉 = 〈(oc‘ndx),
(𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉) |
33 | 26, 32 | preq12d 4677 |
. . . . . 6
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
{〈(le‘ndx), 𝑜〉, 〈(oc‘ndx), (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})〉} = {〈(le‘ndx),
≤
〉, 〈(oc‘ndx), (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}) |
34 | 25, 33 | uneq12d 4097 |
. . . . 5
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
({〈(Base‘ndx), 𝑓〉, 〈(TopSet‘ndx),
(ordTop‘𝑜)〉}
∪ {〈(le‘ndx), 𝑜〉, 〈(oc‘ndx), (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})〉}) =
({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx),
(ordTop‘ ≤ )〉} ∪
{〈(le‘ndx), ≤ 〉,
〈(oc‘ndx), (𝑥
∈ 𝐹 ↦ ∪ {𝑦
∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉})) |
35 | 14, 19, 34 | csbied2 3871 |
. . . 4
⊢ (𝑓 = 𝐹 → ⦋{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} / 𝑜⦌({〈(Base‘ndx),
𝑓〉,
〈(TopSet‘ndx), (ordTop‘𝑜)〉} ∪ {〈(le‘ndx), 𝑜〉, 〈(oc‘ndx),
(𝑥 ∈ 𝑓 ↦ ∪ {𝑦
∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})〉}) =
({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx),
(ordTop‘ ≤ )〉} ∪
{〈(le‘ndx), ≤ 〉,
〈(oc‘ndx), (𝑥
∈ 𝐹 ↦ ∪ {𝑦
∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉})) |
36 | | df-ipo 18256 |
. . . 4
⊢ toInc =
(𝑓 ∈ V ↦
⦋{〈𝑥,
𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} / 𝑜⦌({〈(Base‘ndx),
𝑓〉,
〈(TopSet‘ndx), (ordTop‘𝑜)〉} ∪ {〈(le‘ndx), 𝑜〉, 〈(oc‘ndx),
(𝑥 ∈ 𝑓 ↦ ∪ {𝑦
∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})〉})) |
37 | | prex 5353 |
. . . . 5
⊢
{〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx),
(ordTop‘ ≤ )〉} ∈
V |
38 | | prex 5353 |
. . . . 5
⊢
{〈(le‘ndx), ≤ 〉,
〈(oc‘ndx), (𝑥
∈ 𝐹 ↦ ∪ {𝑦
∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉} ∈
V |
39 | 37, 38 | unex 7586 |
. . . 4
⊢
({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx),
(ordTop‘ ≤ )〉} ∪
{〈(le‘ndx), ≤ 〉,
〈(oc‘ndx), (𝑥
∈ 𝐹 ↦ ∪ {𝑦
∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}) ∈
V |
40 | 35, 36, 39 | fvmpt 6867 |
. . 3
⊢ (𝐹 ∈ V →
(toInc‘𝐹) =
({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx),
(ordTop‘ ≤ )〉} ∪
{〈(le‘ndx), ≤ 〉,
〈(oc‘ndx), (𝑥
∈ 𝐹 ↦ ∪ {𝑦
∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉})) |
41 | 2, 40 | eqtrid 2790 |
. 2
⊢ (𝐹 ∈ V → 𝐼 = ({〈(Base‘ndx),
𝐹〉,
〈(TopSet‘ndx), (ordTop‘ ≤ )〉} ∪
{〈(le‘ndx), ≤ 〉,
〈(oc‘ndx), (𝑥
∈ 𝐹 ↦ ∪ {𝑦
∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉})) |
42 | 1, 41 | syl 17 |
1
⊢ (𝐹 ∈ 𝑉 → 𝐼 = ({〈(Base‘ndx), 𝐹〉,
〈(TopSet‘ndx), (ordTop‘ ≤ )〉} ∪
{〈(le‘ndx), ≤ 〉,
〈(oc‘ndx), (𝑥
∈ 𝐹 ↦ ∪ {𝑦
∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉})) |