| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | elex 3500 | . 2
⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | 
| 2 |  | ipoval.i | . . 3
⊢ 𝐼 = (toInc‘𝐹) | 
| 3 |  | vex 3483 | . . . . . . . 8
⊢ 𝑓 ∈ V | 
| 4 | 3, 3 | xpex 7774 | . . . . . . 7
⊢ (𝑓 × 𝑓) ∈ V | 
| 5 |  | simpl 482 | . . . . . . . . . 10
⊢ (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) → {𝑥, 𝑦} ⊆ 𝑓) | 
| 6 |  | vex 3483 | . . . . . . . . . . 11
⊢ 𝑥 ∈ V | 
| 7 |  | vex 3483 | . . . . . . . . . . 11
⊢ 𝑦 ∈ V | 
| 8 | 6, 7 | prss 4819 | . . . . . . . . . 10
⊢ ((𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓) ↔ {𝑥, 𝑦} ⊆ 𝑓) | 
| 9 | 5, 8 | sylibr 234 | . . . . . . . . 9
⊢ (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) → (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)) | 
| 10 | 9 | ssopab2i 5554 | . . . . . . . 8
⊢
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)} | 
| 11 |  | df-xp 5690 | . . . . . . . 8
⊢ (𝑓 × 𝑓) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑓 ∧ 𝑦 ∈ 𝑓)} | 
| 12 | 10, 11 | sseqtrri 4032 | . . . . . . 7
⊢
{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} ⊆ (𝑓 × 𝑓) | 
| 13 | 4, 12 | ssexi 5321 | . . . . . 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 4009 | . . . . . . . 8
⊢ (𝑓 = 𝐹 → ({𝑥, 𝑦} ⊆ 𝑓 ↔ {𝑥, 𝑦} ⊆ 𝐹)) | 
| 16 | 15 | anbi1d 631 | . . . . . . 7
⊢ (𝑓 = 𝐹 → (({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝐹 ∧ 𝑥 ⊆ 𝑦))) | 
| 17 | 16 | opabbidv 5208 | . . . . . 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 2794 | . . . . 5
⊢ (𝑓 = 𝐹 → {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} = ≤ ) | 
| 20 |  | simpl 482 | . . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → 𝑓 = 𝐹) | 
| 21 | 20 | opeq2d 4879 | . . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
〈(Base‘ndx), 𝑓〉 = 〈(Base‘ndx), 𝐹〉) | 
| 22 |  | simpr 484 | . . . . . . . . 9
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → 𝑜 = ≤ ) | 
| 23 | 22 | fveq2d 6909 | . . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
(ordTop‘𝑜) =
(ordTop‘ ≤ )) | 
| 24 | 23 | opeq2d 4879 | . . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
〈(TopSet‘ndx), (ordTop‘𝑜)〉 = 〈(TopSet‘ndx),
(ordTop‘ ≤
)〉) | 
| 25 | 21, 24 | preq12d 4740 | . . . . . 6
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
{〈(Base‘ndx), 𝑓〉, 〈(TopSet‘ndx),
(ordTop‘𝑜)〉} =
{〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx),
(ordTop‘ ≤
)〉}) | 
| 26 | 22 | opeq2d 4879 | . . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
〈(le‘ndx), 𝑜〉 = 〈(le‘ndx), ≤
〉) | 
| 27 |  | id 22 | . . . . . . . . . 10
⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹) | 
| 28 |  | rabeq 3450 | . . . . . . . . . . 11
⊢ (𝑓 = 𝐹 → {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅} = {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅}) | 
| 29 | 28 | unieqd 4919 | . . . . . . . . . 10
⊢ (𝑓 = 𝐹 → ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅} = ∪
{𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅}) | 
| 30 | 27, 29 | mpteq12dv 5232 | . . . . . . . . 9
⊢ (𝑓 = 𝐹 → (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅}) = (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})) | 
| 31 | 30 | adantr 480 | . . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) → (𝑥 ∈ 𝑓 ↦ ∪ {𝑦 ∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅}) = (𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})) | 
| 32 | 31 | opeq2d 4879 | . . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑜 = ≤ ) →
〈(oc‘ndx), (𝑥
∈ 𝑓 ↦ ∪ {𝑦
∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})〉 = 〈(oc‘ndx),
(𝑥 ∈ 𝐹 ↦ ∪ {𝑦 ∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉) | 
| 33 | 26, 32 | preq12d 4740 | . . . . . 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 4168 | . . . . 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 3935 | . . . 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 18574 | . . . 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 5436 | . . . . 5
⊢
{〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx),
(ordTop‘ ≤ )〉} ∈
V | 
| 38 |  | prex 5436 | . . . . 5
⊢
{〈(le‘ndx), ≤ 〉,
〈(oc‘ndx), (𝑥
∈ 𝐹 ↦ ∪ {𝑦
∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉} ∈
V | 
| 39 | 37, 38 | unex 7765 | . . . 4
⊢
({〈(Base‘ndx), 𝐹〉, 〈(TopSet‘ndx),
(ordTop‘ ≤ )〉} ∪
{〈(le‘ndx), ≤ 〉,
〈(oc‘ndx), (𝑥
∈ 𝐹 ↦ ∪ {𝑦
∈ 𝐹 ∣ (𝑦 ∩ 𝑥) = ∅})〉}) ∈
V | 
| 40 | 35, 36, 39 | fvmpt 7015 | . . 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 2788 | . 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) = ∅})〉})) |