Step | Hyp | Ref
| Expression |
1 | | reltpos 7899 |
. 2
⊢ Rel tpos
tpos 𝐹 |
2 | | relinxp 5689 |
. 2
⊢ Rel
(𝐹 ∩ (((V × V)
∪ {∅}) × V)) |
3 | | relcnv 5969 |
. . . . . . . . 9
⊢ Rel ◡dom tpos 𝐹 |
4 | | df-rel 5564 |
. . . . . . . . 9
⊢ (Rel
◡dom tpos 𝐹 ↔ ◡dom tpos 𝐹 ⊆ (V × V)) |
5 | 3, 4 | mpbi 232 |
. . . . . . . 8
⊢ ◡dom tpos 𝐹 ⊆ (V × V) |
6 | | simpl 485 |
. . . . . . . 8
⊢ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) → 𝑤 ∈ ◡dom tpos 𝐹) |
7 | 5, 6 | sseldi 3967 |
. . . . . . 7
⊢ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) → 𝑤 ∈ (V × V)) |
8 | | simpr 487 |
. . . . . . 7
⊢ ((𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)) → 𝑤 ∈ (V ×
V)) |
9 | | elvv 5628 |
. . . . . . . . 9
⊢ (𝑤 ∈ (V × V) ↔
∃𝑥∃𝑦 𝑤 = 〈𝑥, 𝑦〉) |
10 | | eleq1 2902 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑤 ∈ ◡dom tpos 𝐹 ↔ 〈𝑥, 𝑦〉 ∈ ◡dom tpos 𝐹)) |
11 | | vex 3499 |
. . . . . . . . . . . . . . 15
⊢ 𝑥 ∈ V |
12 | | vex 3499 |
. . . . . . . . . . . . . . 15
⊢ 𝑦 ∈ V |
13 | 11, 12 | opelcnv 5754 |
. . . . . . . . . . . . . 14
⊢
(〈𝑥, 𝑦〉 ∈ ◡dom tpos 𝐹 ↔ 〈𝑦, 𝑥〉 ∈ dom tpos 𝐹) |
14 | 10, 13 | syl6bb 289 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑤 ∈ ◡dom tpos 𝐹 ↔ 〈𝑦, 𝑥〉 ∈ dom tpos 𝐹)) |
15 | | sneq 4579 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 = 〈𝑥, 𝑦〉 → {𝑤} = {〈𝑥, 𝑦〉}) |
16 | 15 | cnveqd 5748 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 〈𝑥, 𝑦〉 → ◡{𝑤} = ◡{〈𝑥, 𝑦〉}) |
17 | 16 | unieqd 4854 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 〈𝑥, 𝑦〉 → ∪
◡{𝑤} = ∪ ◡{〈𝑥, 𝑦〉}) |
18 | | opswap 6088 |
. . . . . . . . . . . . . . 15
⊢ ∪ ◡{〈𝑥, 𝑦〉} = 〈𝑦, 𝑥〉 |
19 | 17, 18 | syl6eq 2874 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 〈𝑥, 𝑦〉 → ∪
◡{𝑤} = 〈𝑦, 𝑥〉) |
20 | 19 | breq1d 5078 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 〈𝑥, 𝑦〉 → (∪
◡{𝑤}tpos 𝐹𝑧 ↔ 〈𝑦, 𝑥〉tpos 𝐹𝑧)) |
21 | 14, 20 | anbi12d 632 |
. . . . . . . . . . . 12
⊢ (𝑤 = 〈𝑥, 𝑦〉 → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (〈𝑦, 𝑥〉 ∈ dom tpos 𝐹 ∧ 〈𝑦, 𝑥〉tpos 𝐹𝑧))) |
22 | | opex 5358 |
. . . . . . . . . . . . . . 15
⊢
〈𝑦, 𝑥〉 ∈ V |
23 | | vex 3499 |
. . . . . . . . . . . . . . 15
⊢ 𝑧 ∈ V |
24 | 22, 23 | breldm 5779 |
. . . . . . . . . . . . . 14
⊢
(〈𝑦, 𝑥〉tpos 𝐹𝑧 → 〈𝑦, 𝑥〉 ∈ dom tpos 𝐹) |
25 | 24 | pm4.71ri 563 |
. . . . . . . . . . . . 13
⊢
(〈𝑦, 𝑥〉tpos 𝐹𝑧 ↔ (〈𝑦, 𝑥〉 ∈ dom tpos 𝐹 ∧ 〈𝑦, 𝑥〉tpos 𝐹𝑧)) |
26 | | brtpos 7903 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ V → (〈𝑦, 𝑥〉tpos 𝐹𝑧 ↔ 〈𝑥, 𝑦〉𝐹𝑧)) |
27 | 26 | elv 3501 |
. . . . . . . . . . . . 13
⊢
(〈𝑦, 𝑥〉tpos 𝐹𝑧 ↔ 〈𝑥, 𝑦〉𝐹𝑧) |
28 | 25, 27 | bitr3i 279 |
. . . . . . . . . . . 12
⊢
((〈𝑦, 𝑥〉 ∈ dom tpos 𝐹 ∧ 〈𝑦, 𝑥〉tpos 𝐹𝑧) ↔ 〈𝑥, 𝑦〉𝐹𝑧) |
29 | 21, 28 | syl6bb 289 |
. . . . . . . . . . 11
⊢ (𝑤 = 〈𝑥, 𝑦〉 → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ 〈𝑥, 𝑦〉𝐹𝑧)) |
30 | | breq1 5071 |
. . . . . . . . . . 11
⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑤𝐹𝑧 ↔ 〈𝑥, 𝑦〉𝐹𝑧)) |
31 | 29, 30 | bitr4d 284 |
. . . . . . . . . 10
⊢ (𝑤 = 〈𝑥, 𝑦〉 → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧)) |
32 | 31 | exlimivv 1933 |
. . . . . . . . 9
⊢
(∃𝑥∃𝑦 𝑤 = 〈𝑥, 𝑦〉 → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧)) |
33 | 9, 32 | sylbi 219 |
. . . . . . . 8
⊢ (𝑤 ∈ (V × V) →
((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧)) |
34 | | iba 530 |
. . . . . . . 8
⊢ (𝑤 ∈ (V × V) →
(𝑤𝐹𝑧 ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)))) |
35 | 33, 34 | bitrd 281 |
. . . . . . 7
⊢ (𝑤 ∈ (V × V) →
((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)))) |
36 | 7, 8, 35 | pm5.21nii 382 |
. . . . . 6
⊢ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V))) |
37 | | elsni 4586 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ∈ {∅} → 𝑤 = ∅) |
38 | 37 | sneqd 4581 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 ∈ {∅} → {𝑤} = {∅}) |
39 | 38 | cnveqd 5748 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ∈ {∅} → ◡{𝑤} = ◡{∅}) |
40 | | cnvsn0 6069 |
. . . . . . . . . . . . . 14
⊢ ◡{∅} = ∅ |
41 | 39, 40 | syl6eq 2874 |
. . . . . . . . . . . . 13
⊢ (𝑤 ∈ {∅} → ◡{𝑤} = ∅) |
42 | 41 | unieqd 4854 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ {∅} → ∪ ◡{𝑤} = ∪
∅) |
43 | | uni0 4868 |
. . . . . . . . . . . 12
⊢ ∪ ∅ = ∅ |
44 | 42, 43 | syl6eq 2874 |
. . . . . . . . . . 11
⊢ (𝑤 ∈ {∅} → ∪ ◡{𝑤} = ∅) |
45 | 44 | breq1d 5078 |
. . . . . . . . . 10
⊢ (𝑤 ∈ {∅} → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ ∅tpos 𝐹𝑧)) |
46 | | brtpos0 7901 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ V → (∅tpos
𝐹𝑧 ↔ ∅𝐹𝑧)) |
47 | 46 | elv 3501 |
. . . . . . . . . 10
⊢
(∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧) |
48 | 45, 47 | syl6bb 289 |
. . . . . . . . 9
⊢ (𝑤 ∈ {∅} → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ ∅𝐹𝑧)) |
49 | 37 | breq1d 5078 |
. . . . . . . . 9
⊢ (𝑤 ∈ {∅} → (𝑤𝐹𝑧 ↔ ∅𝐹𝑧)) |
50 | 48, 49 | bitr4d 284 |
. . . . . . . 8
⊢ (𝑤 ∈ {∅} → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ 𝑤𝐹𝑧)) |
51 | 50 | pm5.32i 577 |
. . . . . . 7
⊢ ((𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤 ∈ {∅} ∧ 𝑤𝐹𝑧)) |
52 | 51 | biancomi 465 |
. . . . . 6
⊢ ((𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ {∅})) |
53 | 36, 52 | orbi12i 911 |
. . . . 5
⊢ (((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧)) ↔ ((𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)) ∨ (𝑤𝐹𝑧 ∧ 𝑤 ∈ {∅}))) |
54 | | andir 1005 |
. . . . 5
⊢ (((𝑤 ∈ ◡dom tpos 𝐹 ∨ 𝑤 ∈ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧))) |
55 | | andi 1004 |
. . . . 5
⊢ ((𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})) ↔ ((𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)) ∨ (𝑤𝐹𝑧 ∧ 𝑤 ∈ {∅}))) |
56 | 53, 54, 55 | 3bitr4i 305 |
. . . 4
⊢ (((𝑤 ∈ ◡dom tpos 𝐹 ∨ 𝑤 ∈ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))) |
57 | | elun 4127 |
. . . . 5
⊢ (𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ↔ (𝑤 ∈ ◡dom tpos 𝐹 ∨ 𝑤 ∈ {∅})) |
58 | 57 | anbi1i 625 |
. . . 4
⊢ ((𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ ((𝑤 ∈ ◡dom tpos 𝐹 ∨ 𝑤 ∈ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧)) |
59 | | brxp 5603 |
. . . . . . 7
⊢ (𝑤(((V × V) ∪ {∅})
× V)𝑧 ↔ (𝑤 ∈ ((V × V) ∪
{∅}) ∧ 𝑧 ∈
V)) |
60 | 23, 59 | mpbiran2 708 |
. . . . . 6
⊢ (𝑤(((V × V) ∪ {∅})
× V)𝑧 ↔ 𝑤 ∈ ((V × V) ∪
{∅})) |
61 | | elun 4127 |
. . . . . 6
⊢ (𝑤 ∈ ((V × V) ∪
{∅}) ↔ (𝑤 ∈
(V × V) ∨ 𝑤 ∈
{∅})) |
62 | 60, 61 | bitri 277 |
. . . . 5
⊢ (𝑤(((V × V) ∪ {∅})
× V)𝑧 ↔ (𝑤 ∈ (V × V) ∨ 𝑤 ∈
{∅})) |
63 | 62 | anbi2i 624 |
. . . 4
⊢ ((𝑤𝐹𝑧 ∧ 𝑤(((V × V) ∪ {∅}) ×
V)𝑧) ↔ (𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))) |
64 | 56, 58, 63 | 3bitr4i 305 |
. . 3
⊢ ((𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤(((V × V) ∪ {∅}) ×
V)𝑧)) |
65 | | brtpos2 7900 |
. . . 4
⊢ (𝑧 ∈ V → (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧))) |
66 | 65 | elv 3501 |
. . 3
⊢ (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧)) |
67 | | brin 5120 |
. . 3
⊢ (𝑤(𝐹 ∩ (((V × V) ∪ {∅})
× V))𝑧 ↔ (𝑤𝐹𝑧 ∧ 𝑤(((V × V) ∪ {∅}) ×
V)𝑧)) |
68 | 64, 66, 67 | 3bitr4i 305 |
. 2
⊢ (𝑤tpos tpos 𝐹𝑧 ↔ 𝑤(𝐹 ∩ (((V × V) ∪ {∅})
× V))𝑧) |
69 | 1, 2, 68 | eqbrriv 5666 |
1
⊢ tpos tpos
𝐹 = (𝐹 ∩ (((V × V) ∪ {∅})
× V)) |