Step | Hyp | Ref
| Expression |
1 | | ssun1 4086 |
. . . . . 6
⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) |
2 | | ssun2 4087 |
. . . . . . 7
⊢ (𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵)) |
3 | | ordtval.1 |
. . . . . . . . . 10
⊢ 𝑋 = dom 𝑅 |
4 | | ordtval.2 |
. . . . . . . . . 10
⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) |
5 | | ordtval.3 |
. . . . . . . . . 10
⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) |
6 | 3, 4, 5 | ordtuni 22087 |
. . . . . . . . 9
⊢ (𝑅 ∈ TosetRel → 𝑋 = ∪
({𝑋} ∪ (𝐴 ∪ 𝐵))) |
7 | | dmexg 7681 |
. . . . . . . . . 10
⊢ (𝑅 ∈ TosetRel → dom
𝑅 ∈
V) |
8 | 3, 7 | eqeltrid 2842 |
. . . . . . . . 9
⊢ (𝑅 ∈ TosetRel → 𝑋 ∈ V) |
9 | 6, 8 | eqeltrrd 2839 |
. . . . . . . 8
⊢ (𝑅 ∈ TosetRel → ∪ ({𝑋}
∪ (𝐴 ∪ 𝐵)) ∈ V) |
10 | | uniexb 7549 |
. . . . . . . 8
⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V ↔ ∪ ({𝑋}
∪ (𝐴 ∪ 𝐵)) ∈ V) |
11 | 9, 10 | sylibr 237 |
. . . . . . 7
⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V) |
12 | | ssexg 5216 |
. . . . . . 7
⊢ (((𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∧ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V) → (𝐴 ∪ 𝐵) ∈ V) |
13 | 2, 11, 12 | sylancr 590 |
. . . . . 6
⊢ (𝑅 ∈ TosetRel → (𝐴 ∪ 𝐵) ∈ V) |
14 | | ssexg 5216 |
. . . . . 6
⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V) |
15 | 1, 13, 14 | sylancr 590 |
. . . . 5
⊢ (𝑅 ∈ TosetRel → 𝐴 ∈ V) |
16 | | ssun2 4087 |
. . . . . 6
⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) |
17 | | ssexg 5216 |
. . . . . 6
⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V) |
18 | 16, 13, 17 | sylancr 590 |
. . . . 5
⊢ (𝑅 ∈ TosetRel → 𝐵 ∈ V) |
19 | | elfiun 9046 |
. . . . 5
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) ↔ (𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛)))) |
20 | 15, 18, 19 | syl2anc 587 |
. . . 4
⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) ↔ (𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛)))) |
21 | 3, 4 | ordtbaslem 22085 |
. . . . . . . 8
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐴) = 𝐴) |
22 | 21, 1 | eqsstrdi 3955 |
. . . . . . 7
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐴) ⊆ (𝐴 ∪ 𝐵)) |
23 | | ssun1 4086 |
. . . . . . 7
⊢ (𝐴 ∪ 𝐵) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶) |
24 | 22, 23 | sstrdi 3913 |
. . . . . 6
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐴) ⊆
((𝐴 ∪ 𝐵) ∪ 𝐶)) |
25 | 24 | sseld 3900 |
. . . . 5
⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐴) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
26 | | cnvtsr 18094 |
. . . . . . . . . 10
⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel ) |
27 | | df-rn 5562 |
. . . . . . . . . . 11
⊢ ran 𝑅 = dom ◡𝑅 |
28 | | eqid 2737 |
. . . . . . . . . . 11
⊢ ran
(𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) |
29 | 27, 28 | ordtbaslem 22085 |
. . . . . . . . . 10
⊢ (◡𝑅 ∈ TosetRel → (fi‘ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) |
30 | 26, 29 | syl 17 |
. . . . . . . . 9
⊢ (𝑅 ∈ TosetRel →
(fi‘ran (𝑥 ∈ ran
𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) |
31 | | tsrps 18093 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ TosetRel → 𝑅 ∈
PosetRel) |
32 | 3 | psrn 18081 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅) |
33 | 31, 32 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ TosetRel → 𝑋 = ran 𝑅) |
34 | | vex 3412 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑦 ∈ V |
35 | | vex 3412 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑥 ∈ V |
36 | 34, 35 | brcnv 5751 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
37 | 36 | bicomi 227 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥𝑅𝑦 ↔ 𝑦◡𝑅𝑥) |
38 | 37 | notbii 323 |
. . . . . . . . . . . . . . 15
⊢ (¬
𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥) |
39 | 38 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ TosetRel → (¬
𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥)) |
40 | 33, 39 | rabeqbidv 3396 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ TosetRel → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) |
41 | 33, 40 | mpteq12dv 5140 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ TosetRel → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) |
42 | 41 | rneqd 5807 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ TosetRel → ran
(𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) |
43 | 5, 42 | syl5eq 2790 |
. . . . . . . . . 10
⊢ (𝑅 ∈ TosetRel → 𝐵 = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) |
44 | 43 | fveq2d 6721 |
. . . . . . . . 9
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) =
(fi‘ran (𝑥 ∈ ran
𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}))) |
45 | 30, 44, 43 | 3eqtr4d 2787 |
. . . . . . . 8
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) = 𝐵) |
46 | 45, 16 | eqsstrdi 3955 |
. . . . . . 7
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) ⊆ (𝐴 ∪ 𝐵)) |
47 | 46, 23 | sstrdi 3913 |
. . . . . 6
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) ⊆
((𝐴 ∪ 𝐵) ∪ 𝐶)) |
48 | 47 | sseld 3900 |
. . . . 5
⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐵) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
49 | | ssun2 4087 |
. . . . . . . 8
⊢ 𝐶 ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶) |
50 | 21, 4 | eqtrdi 2794 |
. . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐴) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})) |
51 | 50 | eleq2d 2823 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ 𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) |
52 | | breq2 5057 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑎 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑎)) |
53 | 52 | notbid 321 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎)) |
54 | 53 | rabbidv 3390 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) |
55 | 54 | cbvmptv 5158 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) |
56 | 55 | elrnmpt 5825 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ V → (𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})) |
57 | 56 | elv 3414 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) |
58 | 51, 57 | bitrdi 290 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})) |
59 | 45, 5 | eqtrdi 2794 |
. . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})) |
60 | 59 | eleq2d 2823 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ 𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))) |
61 | | breq1 5056 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑏 → (𝑥𝑅𝑦 ↔ 𝑏𝑅𝑦)) |
62 | 61 | notbid 321 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑏 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑏𝑅𝑦)) |
63 | 62 | rabbidv 3390 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑏 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) |
64 | 63 | cbvmptv 5158 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) |
65 | 64 | elrnmpt 5825 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ V → (𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})) |
66 | 65 | elv 3414 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) |
67 | 60, 66 | bitrdi 290 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})) |
68 | 58, 67 | anbi12d 634 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) ↔ (∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}))) |
69 | | reeanv 3279 |
. . . . . . . . . . . . 13
⊢
(∃𝑎 ∈
𝑋 ∃𝑏 ∈ 𝑋 (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) ↔ (∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})) |
70 | | ineq12 4122 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚 ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})) |
71 | | inrab 4221 |
. . . . . . . . . . . . . . . 16
⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} |
72 | 70, 71 | eqtrdi 2794 |
. . . . . . . . . . . . . . 15
⊢ ((𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
73 | 72 | reximi 3166 |
. . . . . . . . . . . . . 14
⊢
(∃𝑏 ∈
𝑋 (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
74 | 73 | reximi 3166 |
. . . . . . . . . . . . 13
⊢
(∃𝑎 ∈
𝑋 ∃𝑏 ∈ 𝑋 (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
75 | 69, 74 | sylbir 238 |
. . . . . . . . . . . 12
⊢
((∃𝑎 ∈
𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
76 | 68, 75 | syl6bi 256 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})) |
77 | 76 | imp 410 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
78 | | vex 3412 |
. . . . . . . . . . . 12
⊢ 𝑚 ∈ V |
79 | 78 | inex1 5210 |
. . . . . . . . . . 11
⊢ (𝑚 ∩ 𝑛) ∈ V |
80 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
81 | 80 | elrnmpog 7345 |
. . . . . . . . . . 11
⊢ ((𝑚 ∩ 𝑛) ∈ V → ((𝑚 ∩ 𝑛) ∈ ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})) |
82 | 79, 81 | ax-mp 5 |
. . . . . . . . . 10
⊢ ((𝑚 ∩ 𝑛) ∈ ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ↔ ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
83 | 77, 82 | sylibr 237 |
. . . . . . . . 9
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})) |
84 | | ordtval.4 |
. . . . . . . . 9
⊢ 𝐶 = ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
85 | 83, 84 | eleqtrrdi 2849 |
. . . . . . . 8
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ 𝐶) |
86 | 49, 85 | sseldi 3899 |
. . . . . . 7
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)) |
87 | | eleq1 2825 |
. . . . . . 7
⊢ (𝑧 = (𝑚 ∩ 𝑛) → (𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶) ↔ (𝑚 ∩ 𝑛) ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
88 | 86, 87 | syl5ibrcom 250 |
. . . . . 6
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
89 | 88 | rexlimdvva 3213 |
. . . . 5
⊢ (𝑅 ∈ TosetRel →
(∃𝑚 ∈
(fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
90 | 25, 48, 89 | 3jaod 1430 |
. . . 4
⊢ (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛)) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
91 | 20, 90 | sylbid 243 |
. . 3
⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
92 | 91 | ssrdv 3907 |
. 2
⊢ (𝑅 ∈ TosetRel →
(fi‘(𝐴 ∪ 𝐵)) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶)) |
93 | | ssfii 9035 |
. . . 4
⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵))) |
94 | 13, 93 | syl 17 |
. . 3
⊢ (𝑅 ∈ TosetRel → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵))) |
95 | 94 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵))) |
96 | | simprl 771 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 ∈ 𝑋) |
97 | | eqidd 2738 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) |
98 | 54 | rspceeqv 3552 |
. . . . . . . . . . . . . 14
⊢ ((𝑎 ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) |
99 | 96, 97, 98 | syl2anc 587 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) |
100 | 8 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑋 ∈ V) |
101 | | rabexg 5224 |
. . . . . . . . . . . . . 14
⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V) |
102 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) |
103 | 102 | elrnmpt 5825 |
. . . . . . . . . . . . . 14
⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})) |
104 | 100, 101,
103 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})) |
105 | 99, 104 | mpbird 260 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})) |
106 | 105, 4 | eleqtrrdi 2849 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ 𝐴) |
107 | 1, 106 | sseldi 3899 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (𝐴 ∪ 𝐵)) |
108 | 95, 107 | sseldd 3902 |
. . . . . . . . 9
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴 ∪ 𝐵))) |
109 | | simprr 773 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑏 ∈ 𝑋) |
110 | | eqidd 2738 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) |
111 | 63 | rspceeqv 3552 |
. . . . . . . . . . . . . 14
⊢ ((𝑏 ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) |
112 | 109, 110,
111 | syl2anc 587 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) |
113 | | rabexg 5224 |
. . . . . . . . . . . . . 14
⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V) |
114 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) |
115 | 114 | elrnmpt 5825 |
. . . . . . . . . . . . . 14
⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})) |
116 | 100, 113,
115 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})) |
117 | 112, 116 | mpbird 260 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})) |
118 | 117, 5 | eleqtrrdi 2849 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ 𝐵) |
119 | 16, 118 | sseldi 3899 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (𝐴 ∪ 𝐵)) |
120 | 95, 119 | sseldd 3902 |
. . . . . . . . 9
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴 ∪ 𝐵))) |
121 | | fiin 9038 |
. . . . . . . . 9
⊢ (({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴 ∪ 𝐵)) ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴 ∪ 𝐵))) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴 ∪ 𝐵))) |
122 | 108, 120,
121 | syl2anc 587 |
. . . . . . . 8
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴 ∪ 𝐵))) |
123 | 71, 122 | eqeltrrid 2843 |
. . . . . . 7
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴 ∪ 𝐵))) |
124 | 123 | ralrimivva 3112 |
. . . . . 6
⊢ (𝑅 ∈ TosetRel →
∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴 ∪ 𝐵))) |
125 | 80 | fmpo 7838 |
. . . . . 6
⊢
(∀𝑎 ∈
𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴 ∪ 𝐵)) ↔ (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴 ∪ 𝐵))) |
126 | 124, 125 | sylib 221 |
. . . . 5
⊢ (𝑅 ∈ TosetRel → (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴 ∪ 𝐵))) |
127 | 126 | frnd 6553 |
. . . 4
⊢ (𝑅 ∈ TosetRel → ran
(𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ⊆ (fi‘(𝐴 ∪ 𝐵))) |
128 | 84, 127 | eqsstrid 3949 |
. . 3
⊢ (𝑅 ∈ TosetRel → 𝐶 ⊆ (fi‘(𝐴 ∪ 𝐵))) |
129 | 94, 128 | unssd 4100 |
. 2
⊢ (𝑅 ∈ TosetRel → ((𝐴 ∪ 𝐵) ∪ 𝐶) ⊆ (fi‘(𝐴 ∪ 𝐵))) |
130 | 92, 129 | eqssd 3918 |
1
⊢ (𝑅 ∈ TosetRel →
(fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶)) |