Step | Hyp | Ref
| Expression |
1 | | reltpos 6226 |
. 2
⊢ Rel tpos
tpos 𝐹 |
2 | | inss2 3348 |
. . 3
⊢ (𝐹 ∩ (((V × V) ∪
{∅}) × V)) ⊆ (((V × V) ∪ {∅}) ×
V) |
3 | | relxp 4718 |
. . 3
⊢ Rel (((V
× V) ∪ {∅}) × V) |
4 | | relss 4696 |
. . 3
⊢ ((𝐹 ∩ (((V × V) ∪
{∅}) × V)) ⊆ (((V × V) ∪ {∅}) × V)
→ (Rel (((V × V) ∪ {∅}) × V) → Rel (𝐹 ∩ (((V × V) ∪
{∅}) × V)))) |
5 | 2, 3, 4 | mp2 16 |
. 2
⊢ Rel
(𝐹 ∩ (((V × V)
∪ {∅}) × V)) |
6 | | relcnv 4987 |
. . . . . . . . 9
⊢ Rel ◡dom tpos 𝐹 |
7 | | df-rel 4616 |
. . . . . . . . 9
⊢ (Rel
◡dom tpos 𝐹 ↔ ◡dom tpos 𝐹 ⊆ (V × V)) |
8 | 6, 7 | mpbi 144 |
. . . . . . . 8
⊢ ◡dom tpos 𝐹 ⊆ (V × V) |
9 | | simpl 108 |
. . . . . . . 8
⊢ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) → 𝑤 ∈ ◡dom tpos 𝐹) |
10 | 8, 9 | sselid 3145 |
. . . . . . 7
⊢ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) → 𝑤 ∈ (V × V)) |
11 | | simpr 109 |
. . . . . . 7
⊢ ((𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)) → 𝑤 ∈ (V ×
V)) |
12 | | elvv 4671 |
. . . . . . . . 9
⊢ (𝑤 ∈ (V × V) ↔
∃𝑥∃𝑦 𝑤 = 〈𝑥, 𝑦〉) |
13 | | eleq1 2233 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑤 ∈ ◡dom tpos 𝐹 ↔ 〈𝑥, 𝑦〉 ∈ ◡dom tpos 𝐹)) |
14 | | vex 2733 |
. . . . . . . . . . . . . . 15
⊢ 𝑥 ∈ V |
15 | | vex 2733 |
. . . . . . . . . . . . . . 15
⊢ 𝑦 ∈ V |
16 | 14, 15 | opelcnv 4791 |
. . . . . . . . . . . . . 14
⊢
(〈𝑥, 𝑦〉 ∈ ◡dom tpos 𝐹 ↔ 〈𝑦, 𝑥〉 ∈ dom tpos 𝐹) |
17 | 13, 16 | bitrdi 195 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑤 ∈ ◡dom tpos 𝐹 ↔ 〈𝑦, 𝑥〉 ∈ dom tpos 𝐹)) |
18 | | sneq 3592 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 = 〈𝑥, 𝑦〉 → {𝑤} = {〈𝑥, 𝑦〉}) |
19 | 18 | cnveqd 4785 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 〈𝑥, 𝑦〉 → ◡{𝑤} = ◡{〈𝑥, 𝑦〉}) |
20 | 19 | unieqd 3805 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 〈𝑥, 𝑦〉 → ∪
◡{𝑤} = ∪ ◡{〈𝑥, 𝑦〉}) |
21 | | opswapg 5095 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → ∪ ◡{〈𝑥, 𝑦〉} = 〈𝑦, 𝑥〉) |
22 | 14, 15, 21 | mp2an 424 |
. . . . . . . . . . . . . . 15
⊢ ∪ ◡{〈𝑥, 𝑦〉} = 〈𝑦, 𝑥〉 |
23 | 20, 22 | eqtrdi 2219 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 〈𝑥, 𝑦〉 → ∪
◡{𝑤} = 〈𝑦, 𝑥〉) |
24 | 23 | breq1d 3997 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 〈𝑥, 𝑦〉 → (∪
◡{𝑤}tpos 𝐹𝑧 ↔ 〈𝑦, 𝑥〉tpos 𝐹𝑧)) |
25 | 17, 24 | anbi12d 470 |
. . . . . . . . . . . 12
⊢ (𝑤 = 〈𝑥, 𝑦〉 → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (〈𝑦, 𝑥〉 ∈ dom tpos 𝐹 ∧ 〈𝑦, 𝑥〉tpos 𝐹𝑧))) |
26 | 15, 14 | opex 4212 |
. . . . . . . . . . . . . . 15
⊢
〈𝑦, 𝑥〉 ∈ V |
27 | | vex 2733 |
. . . . . . . . . . . . . . 15
⊢ 𝑧 ∈ V |
28 | 26, 27 | breldm 4813 |
. . . . . . . . . . . . . 14
⊢
(〈𝑦, 𝑥〉tpos 𝐹𝑧 → 〈𝑦, 𝑥〉 ∈ dom tpos 𝐹) |
29 | 28 | pm4.71ri 390 |
. . . . . . . . . . . . 13
⊢
(〈𝑦, 𝑥〉tpos 𝐹𝑧 ↔ (〈𝑦, 𝑥〉 ∈ dom tpos 𝐹 ∧ 〈𝑦, 𝑥〉tpos 𝐹𝑧)) |
30 | | brtposg 6230 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V ∧ 𝑧 ∈ V) → (〈𝑦, 𝑥〉tpos 𝐹𝑧 ↔ 〈𝑥, 𝑦〉𝐹𝑧)) |
31 | 15, 14, 27, 30 | mp3an 1332 |
. . . . . . . . . . . . 13
⊢
(〈𝑦, 𝑥〉tpos 𝐹𝑧 ↔ 〈𝑥, 𝑦〉𝐹𝑧) |
32 | 29, 31 | bitr3i 185 |
. . . . . . . . . . . 12
⊢
((〈𝑦, 𝑥〉 ∈ dom tpos 𝐹 ∧ 〈𝑦, 𝑥〉tpos 𝐹𝑧) ↔ 〈𝑥, 𝑦〉𝐹𝑧) |
33 | 25, 32 | bitrdi 195 |
. . . . . . . . . . 11
⊢ (𝑤 = 〈𝑥, 𝑦〉 → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ 〈𝑥, 𝑦〉𝐹𝑧)) |
34 | | breq1 3990 |
. . . . . . . . . . 11
⊢ (𝑤 = 〈𝑥, 𝑦〉 → (𝑤𝐹𝑧 ↔ 〈𝑥, 𝑦〉𝐹𝑧)) |
35 | 33, 34 | bitr4d 190 |
. . . . . . . . . 10
⊢ (𝑤 = 〈𝑥, 𝑦〉 → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧)) |
36 | 35 | exlimivv 1889 |
. . . . . . . . 9
⊢
(∃𝑥∃𝑦 𝑤 = 〈𝑥, 𝑦〉 → ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧)) |
37 | 12, 36 | sylbi 120 |
. . . . . . . 8
⊢ (𝑤 ∈ (V × V) →
((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ 𝑤𝐹𝑧)) |
38 | | iba 298 |
. . . . . . . 8
⊢ (𝑤 ∈ (V × V) →
(𝑤𝐹𝑧 ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)))) |
39 | 37, 38 | bitrd 187 |
. . . . . . 7
⊢ (𝑤 ∈ (V × V) →
((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)))) |
40 | 10, 11, 39 | pm5.21nii 699 |
. . . . . 6
⊢ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V))) |
41 | | elsni 3599 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ∈ {∅} → 𝑤 = ∅) |
42 | 41 | sneqd 3594 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 ∈ {∅} → {𝑤} = {∅}) |
43 | 42 | cnveqd 4785 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ∈ {∅} → ◡{𝑤} = ◡{∅}) |
44 | | cnvsn0 5077 |
. . . . . . . . . . . . . 14
⊢ ◡{∅} = ∅ |
45 | 43, 44 | eqtrdi 2219 |
. . . . . . . . . . . . 13
⊢ (𝑤 ∈ {∅} → ◡{𝑤} = ∅) |
46 | 45 | unieqd 3805 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ {∅} → ∪ ◡{𝑤} = ∪
∅) |
47 | | uni0 3821 |
. . . . . . . . . . . 12
⊢ ∪ ∅ = ∅ |
48 | 46, 47 | eqtrdi 2219 |
. . . . . . . . . . 11
⊢ (𝑤 ∈ {∅} → ∪ ◡{𝑤} = ∅) |
49 | 48 | breq1d 3997 |
. . . . . . . . . 10
⊢ (𝑤 ∈ {∅} → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ ∅tpos 𝐹𝑧)) |
50 | | brtpos0 6228 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ V → (∅tpos
𝐹𝑧 ↔ ∅𝐹𝑧)) |
51 | 27, 50 | ax-mp 5 |
. . . . . . . . . 10
⊢
(∅tpos 𝐹𝑧 ↔ ∅𝐹𝑧) |
52 | 49, 51 | bitrdi 195 |
. . . . . . . . 9
⊢ (𝑤 ∈ {∅} → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ ∅𝐹𝑧)) |
53 | 41 | breq1d 3997 |
. . . . . . . . 9
⊢ (𝑤 ∈ {∅} → (𝑤𝐹𝑧 ↔ ∅𝐹𝑧)) |
54 | 52, 53 | bitr4d 190 |
. . . . . . . 8
⊢ (𝑤 ∈ {∅} → (∪ ◡{𝑤}tpos 𝐹𝑧 ↔ 𝑤𝐹𝑧)) |
55 | 54 | pm5.32i 451 |
. . . . . . 7
⊢ ((𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤 ∈ {∅} ∧ 𝑤𝐹𝑧)) |
56 | | ancom 264 |
. . . . . . 7
⊢ ((𝑤 ∈ {∅} ∧ 𝑤𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ {∅})) |
57 | 55, 56 | bitri 183 |
. . . . . 6
⊢ ((𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤 ∈ {∅})) |
58 | 40, 57 | orbi12i 759 |
. . . . 5
⊢ (((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧)) ↔ ((𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)) ∨ (𝑤𝐹𝑧 ∧ 𝑤 ∈ {∅}))) |
59 | | andir 814 |
. . . . 5
⊢ (((𝑤 ∈ ◡dom tpos 𝐹 ∨ 𝑤 ∈ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ ((𝑤 ∈ ◡dom tpos 𝐹 ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ∨ (𝑤 ∈ {∅} ∧ ∪ ◡{𝑤}tpos 𝐹𝑧))) |
60 | | andi 813 |
. . . . 5
⊢ ((𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅})) ↔ ((𝑤𝐹𝑧 ∧ 𝑤 ∈ (V × V)) ∨ (𝑤𝐹𝑧 ∧ 𝑤 ∈ {∅}))) |
61 | 58, 59, 60 | 3bitr4i 211 |
. . . 4
⊢ (((𝑤 ∈ ◡dom tpos 𝐹 ∨ 𝑤 ∈ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))) |
62 | | elun 3268 |
. . . . 5
⊢ (𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ↔ (𝑤 ∈ ◡dom tpos 𝐹 ∨ 𝑤 ∈ {∅})) |
63 | 62 | anbi1i 455 |
. . . 4
⊢ ((𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ ((𝑤 ∈ ◡dom tpos 𝐹 ∨ 𝑤 ∈ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧)) |
64 | | brxp 4640 |
. . . . . . 7
⊢ (𝑤(((V × V) ∪ {∅})
× V)𝑧 ↔ (𝑤 ∈ ((V × V) ∪
{∅}) ∧ 𝑧 ∈
V)) |
65 | 27, 64 | mpbiran2 936 |
. . . . . 6
⊢ (𝑤(((V × V) ∪ {∅})
× V)𝑧 ↔ 𝑤 ∈ ((V × V) ∪
{∅})) |
66 | | elun 3268 |
. . . . . 6
⊢ (𝑤 ∈ ((V × V) ∪
{∅}) ↔ (𝑤 ∈
(V × V) ∨ 𝑤 ∈
{∅})) |
67 | 65, 66 | bitri 183 |
. . . . 5
⊢ (𝑤(((V × V) ∪ {∅})
× V)𝑧 ↔ (𝑤 ∈ (V × V) ∨ 𝑤 ∈
{∅})) |
68 | 67 | anbi2i 454 |
. . . 4
⊢ ((𝑤𝐹𝑧 ∧ 𝑤(((V × V) ∪ {∅}) ×
V)𝑧) ↔ (𝑤𝐹𝑧 ∧ (𝑤 ∈ (V × V) ∨ 𝑤 ∈ {∅}))) |
69 | 61, 63, 68 | 3bitr4i 211 |
. . 3
⊢ ((𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧) ↔ (𝑤𝐹𝑧 ∧ 𝑤(((V × V) ∪ {∅}) ×
V)𝑧)) |
70 | | brtpos2 6227 |
. . . 4
⊢ (𝑧 ∈ V → (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧))) |
71 | 27, 70 | ax-mp 5 |
. . 3
⊢ (𝑤tpos tpos 𝐹𝑧 ↔ (𝑤 ∈ (◡dom tpos 𝐹 ∪ {∅}) ∧ ∪ ◡{𝑤}tpos 𝐹𝑧)) |
72 | | brin 4039 |
. . 3
⊢ (𝑤(𝐹 ∩ (((V × V) ∪ {∅})
× V))𝑧 ↔ (𝑤𝐹𝑧 ∧ 𝑤(((V × V) ∪ {∅}) ×
V)𝑧)) |
73 | 69, 71, 72 | 3bitr4i 211 |
. 2
⊢ (𝑤tpos tpos 𝐹𝑧 ↔ 𝑤(𝐹 ∩ (((V × V) ∪ {∅})
× V))𝑧) |
74 | 1, 5, 73 | eqbrriv 4704 |
1
⊢ tpos tpos
𝐹 = (𝐹 ∩ (((V × V) ∪ {∅})
× V)) |