| Step | Hyp | Ref
| Expression |
| 1 | | elex 3501 |
. 2
⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V) |
| 2 | | psubspset.s |
. . 3
⊢ 𝑆 = (PSubSp‘𝐾) |
| 3 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾)) |
| 4 | | psubspset.a |
. . . . . . . 8
⊢ 𝐴 = (Atoms‘𝐾) |
| 5 | 3, 4 | eqtr4di 2795 |
. . . . . . 7
⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴) |
| 6 | 5 | sseq2d 4016 |
. . . . . 6
⊢ (𝑘 = 𝐾 → (𝑠 ⊆ (Atoms‘𝑘) ↔ 𝑠 ⊆ 𝐴)) |
| 7 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾)) |
| 8 | | psubspset.j |
. . . . . . . . . . . . 13
⊢ ∨ =
(join‘𝐾) |
| 9 | 7, 8 | eqtr4di 2795 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ ) |
| 10 | 9 | oveqd 7448 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐾 → (𝑝(join‘𝑘)𝑞) = (𝑝 ∨ 𝑞)) |
| 11 | 10 | breq2d 5155 |
. . . . . . . . . 10
⊢ (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟(le‘𝑘)(𝑝 ∨ 𝑞))) |
| 12 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾)) |
| 13 | | psubspset.l |
. . . . . . . . . . . 12
⊢ ≤ =
(le‘𝐾) |
| 14 | 12, 13 | eqtr4di 2795 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ ) |
| 15 | 14 | breqd 5154 |
. . . . . . . . . 10
⊢ (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝 ∨ 𝑞) ↔ 𝑟 ≤ (𝑝 ∨ 𝑞))) |
| 16 | 11, 15 | bitrd 279 |
. . . . . . . . 9
⊢ (𝑘 = 𝐾 → (𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) ↔ 𝑟 ≤ (𝑝 ∨ 𝑞))) |
| 17 | 16 | imbi1d 341 |
. . . . . . . 8
⊢ (𝑘 = 𝐾 → ((𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠) ↔ (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))) |
| 18 | 5, 17 | raleqbidv 3346 |
. . . . . . 7
⊢ (𝑘 = 𝐾 → (∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠) ↔ ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))) |
| 19 | 18 | 2ralbidv 3221 |
. . . . . 6
⊢ (𝑘 = 𝐾 → (∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠) ↔ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))) |
| 20 | 6, 19 | anbi12d 632 |
. . . . 5
⊢ (𝑘 = 𝐾 → ((𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠)) ↔ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠)))) |
| 21 | 20 | abbidv 2808 |
. . . 4
⊢ (𝑘 = 𝐾 → {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠))} = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))}) |
| 22 | | df-psubsp 39505 |
. . . 4
⊢ PSubSp =
(𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠))}) |
| 23 | 4 | fvexi 6920 |
. . . . . 6
⊢ 𝐴 ∈ V |
| 24 | 23 | pwex 5380 |
. . . . 5
⊢ 𝒫
𝐴 ∈ V |
| 25 | | velpw 4605 |
. . . . . . . 8
⊢ (𝑠 ∈ 𝒫 𝐴 ↔ 𝑠 ⊆ 𝐴) |
| 26 | 25 | anbi1i 624 |
. . . . . . 7
⊢ ((𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠)) ↔ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))) |
| 27 | 26 | abbii 2809 |
. . . . . 6
⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} |
| 28 | | ssab2 4079 |
. . . . . 6
⊢ {𝑠 ∣ (𝑠 ∈ 𝒫 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} ⊆ 𝒫 𝐴 |
| 29 | 27, 28 | eqsstrri 4031 |
. . . . 5
⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} ⊆ 𝒫 𝐴 |
| 30 | 24, 29 | ssexi 5322 |
. . . 4
⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))} ∈ V |
| 31 | 21, 22, 30 | fvmpt 7016 |
. . 3
⊢ (𝐾 ∈ V →
(PSubSp‘𝐾) = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))}) |
| 32 | 2, 31 | eqtrid 2789 |
. 2
⊢ (𝐾 ∈ V → 𝑆 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))}) |
| 33 | 1, 32 | syl 17 |
1
⊢ (𝐾 ∈ 𝐵 → 𝑆 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))}) |