Proof of Theorem ordtrest2lem
Step | Hyp | Ref
| Expression |
1 | | inrab2 3859 |
. . . . 5
⊢ ({𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} ∩ 𝐴) = {𝑤 ∈ (𝑋 ∩ 𝐴) ∣ ¬ 𝑤𝑅𝑧} |
2 | | ordtrest2.3 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ⊆ 𝑋) |
3 | | sseqin2 3779 |
. . . . . . . 8
⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∩ 𝐴) = 𝐴) |
4 | 2, 3 | sylib 207 |
. . . . . . 7
⊢ (𝜑 → (𝑋 ∩ 𝐴) = 𝐴) |
5 | 4 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑋 ∩ 𝐴) = 𝐴) |
6 | | rabeq 3166 |
. . . . . 6
⊢ ((𝑋 ∩ 𝐴) = 𝐴 → {𝑤 ∈ (𝑋 ∩ 𝐴) ∣ ¬ 𝑤𝑅𝑧} = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧}) |
7 | 5, 6 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → {𝑤 ∈ (𝑋 ∩ 𝐴) ∣ ¬ 𝑤𝑅𝑧} = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧}) |
8 | 1, 7 | syl5eq 2656 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ({𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} ∩ 𝐴) = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧}) |
9 | | ordtrest2.2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ TosetRel ) |
10 | | inex1g 4729 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V) |
11 | 9, 10 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V) |
12 | | eqid 2610 |
. . . . . . . . . . 11
⊢ dom
(𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴)) |
13 | 12 | ordttopon 20807 |
. . . . . . . . . 10
⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴)))) |
14 | 11, 13 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴)))) |
15 | | tsrps 17044 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ TosetRel → 𝑅 ∈
PosetRel) |
16 | 9, 15 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ PosetRel) |
17 | | ordtrest2.1 |
. . . . . . . . . . . 12
⊢ 𝑋 = dom 𝑅 |
18 | 17 | psssdm 17039 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ⊆ 𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴) |
19 | 16, 2, 18 | syl2anc 691 |
. . . . . . . . . 10
⊢ (𝜑 → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴) |
20 | 19 | fveq2d 6107 |
. . . . . . . . 9
⊢ (𝜑 → (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴)) |
21 | 14, 20 | eleqtrd 2690 |
. . . . . . . 8
⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴)) |
22 | | toponmax 20543 |
. . . . . . . 8
⊢
((ordTop‘(𝑅
∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |
23 | 21, 22 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |
24 | 23 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |
25 | | rabid2 3096 |
. . . . . . 7
⊢ (𝐴 = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ↔ ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧) |
26 | | eleq1 2676 |
. . . . . . 7
⊢ (𝐴 = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} → (𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))) |
27 | 25, 26 | sylbir 224 |
. . . . . 6
⊢
(∀𝑤 ∈
𝐴 ¬ 𝑤𝑅𝑧 → (𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))) |
28 | 24, 27 | syl5ibcom 234 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))) |
29 | | dfrex2 2979 |
. . . . . . 7
⊢
(∃𝑤 ∈
𝐴 𝑤𝑅𝑧 ↔ ¬ ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧) |
30 | | breq1 4586 |
. . . . . . . 8
⊢ (𝑤 = 𝑥 → (𝑤𝑅𝑧 ↔ 𝑥𝑅𝑧)) |
31 | 30 | cbvrexv 3148 |
. . . . . . 7
⊢
(∃𝑤 ∈
𝐴 𝑤𝑅𝑧 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝑧) |
32 | 29, 31 | bitr3i 265 |
. . . . . 6
⊢ (¬
∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝑧) |
33 | | ordttop 20814 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top) |
34 | 11, 33 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top) |
35 | 34 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top) |
36 | | 0opn 20534 |
. . . . . . . . . . 11
⊢
((ordTop‘(𝑅
∩ (𝐴 × 𝐴))) ∈ Top → ∅
∈ (ordTop‘(𝑅
∩ (𝐴 × 𝐴)))) |
37 | 35, 36 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∅ ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |
38 | 37 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → ∅ ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |
39 | | eleq1 2676 |
. . . . . . . . 9
⊢ ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} = ∅ → ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ ∅ ∈
(ordTop‘(𝑅 ∩
(𝐴 × 𝐴))))) |
40 | 38, 39 | syl5ibrcom 236 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} = ∅ → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))) |
41 | | rabn0 3912 |
. . . . . . . . . 10
⊢ ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧) |
42 | | breq1 4586 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑦 → (𝑤𝑅𝑧 ↔ 𝑦𝑅𝑧)) |
43 | 42 | notbid 307 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑦 → (¬ 𝑤𝑅𝑧 ↔ ¬ 𝑦𝑅𝑧)) |
44 | 43 | cbvrexv 3148 |
. . . . . . . . . 10
⊢
(∃𝑤 ∈
𝐴 ¬ 𝑤𝑅𝑧 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) |
45 | 41, 44 | bitri 263 |
. . . . . . . . 9
⊢ ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ≠ ∅ ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧) |
46 | 9 | ad3antrrr 762 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) ∧ 𝑦 ∈ 𝐴) → 𝑅 ∈ TosetRel ) |
47 | 2 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → 𝐴 ⊆ 𝑋) |
48 | 47 | sselda 3568 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝑋) |
49 | | simpllr 795 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑋) |
50 | 17 | tsrlin 17042 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ TosetRel ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦𝑅𝑧 ∨ 𝑧𝑅𝑦)) |
51 | 46, 48, 49, 50 | syl3anc 1318 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑧 ∨ 𝑧𝑅𝑦)) |
52 | 51 | ord 391 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦𝑅𝑧 → 𝑧𝑅𝑦)) |
53 | | an4 861 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧𝑅𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦))) |
54 | | ordtrest2.4 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} ⊆ 𝐴) |
55 | | rabss 3642 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ({𝑧 ∈ 𝑋 ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦)} ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝑋 ((𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦) → 𝑧 ∈ 𝐴)) |
56 | 54, 55 | sylib 207 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ∀𝑧 ∈ 𝑋 ((𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦) → 𝑧 ∈ 𝐴)) |
57 | 56 | r19.21bi 2916 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑧 ∈ 𝑋) → ((𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦) → 𝑧 ∈ 𝐴)) |
58 | 57 | an32s 842 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦) → 𝑧 ∈ 𝐴)) |
59 | 58 | impr 647 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝑅𝑧 ∧ 𝑧𝑅𝑦))) → 𝑧 ∈ 𝐴) |
60 | 53, 59 | sylan2b 491 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) → 𝑧 ∈ 𝐴) |
61 | | brinxp 5104 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑤𝑅𝑧 ↔ 𝑤(𝑅 ∩ (𝐴 × 𝐴))𝑧)) |
62 | 61 | ancoms 468 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑤𝑅𝑧 ↔ 𝑤(𝑅 ∩ (𝐴 × 𝐴))𝑧)) |
63 | 62 | notbid 307 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (¬ 𝑤𝑅𝑧 ↔ ¬ 𝑤(𝑅 ∩ (𝐴 × 𝐴))𝑧)) |
64 | 63 | rabbidva 3163 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∈ 𝐴 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤(𝑅 ∩ (𝐴 × 𝐴))𝑧}) |
65 | 60, 64 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤(𝑅 ∩ (𝐴 × 𝐴))𝑧}) |
66 | 19 | ad2antrr 758 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴) |
67 | | rabeq 3166 |
. . . . . . . . . . . . . . . 16
⊢ (dom
(𝑅 ∩ (𝐴 × 𝐴)) = 𝐴 → {𝑤 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑤(𝑅 ∩ (𝐴 × 𝐴))𝑧} = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤(𝑅 ∩ (𝐴 × 𝐴))𝑧}) |
68 | 66, 67 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) → {𝑤 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑤(𝑅 ∩ (𝐴 × 𝐴))𝑧} = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤(𝑅 ∩ (𝐴 × 𝐴))𝑧}) |
69 | 65, 68 | eqtr4d 2647 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} = {𝑤 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑤(𝑅 ∩ (𝐴 × 𝐴))𝑧}) |
70 | 11 | ad2antrr 758 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V) |
71 | 60, 66 | eleqtrrd 2691 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) → 𝑧 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))) |
72 | 12 | ordtopn1 20808 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∩ (𝐴 × 𝐴)) ∈ V ∧ 𝑧 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))) → {𝑤 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑤(𝑅 ∩ (𝐴 × 𝐴))𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |
73 | 70, 71, 72 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) → {𝑤 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑤(𝑅 ∩ (𝐴 × 𝐴))𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |
74 | 69, 73 | eqeltrd 2688 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧𝑅𝑦))) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |
75 | 74 | anassrs 678 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧𝑅𝑦)) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |
76 | 75 | expr 641 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) ∧ 𝑦 ∈ 𝐴) → (𝑧𝑅𝑦 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))) |
77 | 52, 76 | syld 46 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦𝑅𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))) |
78 | 77 | rexlimdva 3013 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (∃𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))) |
79 | 45, 78 | syl5bi 231 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ≠ ∅ → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))) |
80 | 40, 79 | pm2.61dne 2868 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |
81 | 80 | rexlimdvaa 3014 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (∃𝑥 ∈ 𝐴 𝑥𝑅𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))) |
82 | 32, 81 | syl5bi 231 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (¬ ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))) |
83 | 28, 82 | pm2.61d 169 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤𝑅𝑧} ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |
84 | 8, 83 | eqeltrd 2688 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ({𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |
85 | 84 | ralrimiva 2949 |
. 2
⊢ (𝜑 → ∀𝑧 ∈ 𝑋 ({𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |
86 | | dmexg 6989 |
. . . . . . 7
⊢ (𝑅 ∈ TosetRel → dom
𝑅 ∈
V) |
87 | 9, 86 | syl 17 |
. . . . . 6
⊢ (𝜑 → dom 𝑅 ∈ V) |
88 | 17, 87 | syl5eqel 2692 |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ V) |
89 | | rabexg 4739 |
. . . . 5
⊢ (𝑋 ∈ V → {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} ∈ V) |
90 | 88, 89 | syl 17 |
. . . 4
⊢ (𝜑 → {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} ∈ V) |
91 | 90 | ralrimivw 2950 |
. . 3
⊢ (𝜑 → ∀𝑧 ∈ 𝑋 {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} ∈ V) |
92 | | eqid 2610 |
. . . 4
⊢ (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) = (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧}) |
93 | | ineq1 3769 |
. . . . 5
⊢ (𝑣 = {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} → (𝑣 ∩ 𝐴) = ({𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} ∩ 𝐴)) |
94 | 93 | eleq1d 2672 |
. . . 4
⊢ (𝑣 = {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} → ((𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ ({𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))) |
95 | 92, 94 | ralrnmpt 6276 |
. . 3
⊢
(∀𝑧 ∈
𝑋 {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} ∈ V → (∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ ∀𝑧 ∈ 𝑋 ({𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))) |
96 | 91, 95 | syl 17 |
. 2
⊢ (𝜑 → (∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ ∀𝑧 ∈ 𝑋 ({𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧} ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))) |
97 | 85, 96 | mpbird 246 |
1
⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝑋 ↦ {𝑤 ∈ 𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) |