Step | Hyp | Ref
| Expression |
1 | | relcnv 6060 |
. . . . . . . . . 10
⊢ Rel ◡dom 𝐹 |
2 | | dmtpos 8173 |
. . . . . . . . . . 11
⊢ (Rel dom
𝐹 → dom tpos 𝐹 = ◡dom 𝐹) |
3 | 2 | releqd 5738 |
. . . . . . . . . 10
⊢ (Rel dom
𝐹 → (Rel dom tpos
𝐹 ↔ Rel ◡dom 𝐹)) |
4 | 1, 3 | mpbiri 258 |
. . . . . . . . 9
⊢ (Rel dom
𝐹 → Rel dom tpos 𝐹) |
5 | | reltpos 8166 |
. . . . . . . . 9
⊢ Rel tpos
𝐹 |
6 | 4, 5 | jctil 521 |
. . . . . . . 8
⊢ (Rel dom
𝐹 → (Rel tpos 𝐹 ∧ Rel dom tpos 𝐹)) |
7 | | relrelss 6229 |
. . . . . . . 8
⊢ ((Rel
tpos 𝐹 ∧ Rel dom tpos
𝐹) ↔ tpos 𝐹 ⊆ ((V × V) ×
V)) |
8 | 6, 7 | sylib 217 |
. . . . . . 7
⊢ (Rel dom
𝐹 → tpos 𝐹 ⊆ ((V × V) ×
V)) |
9 | 8 | sseld 3947 |
. . . . . 6
⊢ (Rel dom
𝐹 → (𝑤 ∈ tpos 𝐹 → 𝑤 ∈ ((V × V) ×
V))) |
10 | | elvvv 5711 |
. . . . . 6
⊢ (𝑤 ∈ ((V × V) ×
V) ↔ ∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩) |
11 | 9, 10 | syl6ib 251 |
. . . . 5
⊢ (Rel dom
𝐹 → (𝑤 ∈ tpos 𝐹 → ∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)) |
12 | 11 | pm4.71rd 564 |
. . . 4
⊢ (Rel dom
𝐹 → (𝑤 ∈ tpos 𝐹 ↔ (∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹))) |
13 | | 19.41vvv 1956 |
. . . . 5
⊢
(∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹)) |
14 | | eleq1 2822 |
. . . . . . . 8
⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹)) |
15 | | df-br 5110 |
. . . . . . . . 9
⊢
(⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ tpos 𝐹) |
16 | | brtpos 8170 |
. . . . . . . . . 10
⊢ (𝑧 ∈ V → (⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧)) |
17 | 16 | elv 3453 |
. . . . . . . . 9
⊢
(⟨𝑥, 𝑦⟩tpos 𝐹𝑧 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧) |
18 | 15, 17 | bitr3i 277 |
. . . . . . . 8
⊢
(⟨⟨𝑥,
𝑦⟩, 𝑧⟩ ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧) |
19 | 14, 18 | bitrdi 287 |
. . . . . . 7
⊢ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝑤 ∈ tpos 𝐹 ↔ ⟨𝑦, 𝑥⟩𝐹𝑧)) |
20 | 19 | pm5.32i 576 |
. . . . . 6
⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)) |
21 | 20 | 3exbii 1853 |
. . . . 5
⊢
(∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)) |
22 | 13, 21 | bitr3i 277 |
. . . 4
⊢
((∃𝑥∃𝑦∃𝑧 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝑤 ∈ tpos 𝐹) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)) |
23 | 12, 22 | bitrdi 287 |
. . 3
⊢ (Rel dom
𝐹 → (𝑤 ∈ tpos 𝐹 ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧))) |
24 | 23 | abbi2dv 2868 |
. 2
⊢ (Rel dom
𝐹 → tpos 𝐹 = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)}) |
25 | | df-oprab 7365 |
. 2
⊢
{⟨⟨𝑥,
𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ⟨𝑦, 𝑥⟩𝐹𝑧)} |
26 | 24, 25 | eqtr4di 2791 |
1
⊢ (Rel dom
𝐹 → tpos 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨𝑦, 𝑥⟩𝐹𝑧}) |