Step | Hyp | Ref
| Expression |
1 | | ssun1 3738 |
. . . . . 6
⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) |
2 | | ssun2 3739 |
. . . . . . 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 20804 |
. . . . . . . . 9
⊢ (𝑅 ∈ TosetRel → 𝑋 = ∪
({𝑋} ∪ (𝐴 ∪ 𝐵))) |
7 | | dmexg 6989 |
. . . . . . . . . 10
⊢ (𝑅 ∈ TosetRel → dom
𝑅 ∈
V) |
8 | 3, 7 | syl5eqel 2692 |
. . . . . . . . 9
⊢ (𝑅 ∈ TosetRel → 𝑋 ∈ V) |
9 | 6, 8 | eqeltrrd 2689 |
. . . . . . . 8
⊢ (𝑅 ∈ TosetRel → ∪ ({𝑋}
∪ (𝐴 ∪ 𝐵)) ∈ V) |
10 | | uniexb 6866 |
. . . . . . . 8
⊢ (({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V ↔ ∪ ({𝑋}
∪ (𝐴 ∪ 𝐵)) ∈ V) |
11 | 9, 10 | sylibr 223 |
. . . . . . 7
⊢ (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V) |
12 | | ssexg 4732 |
. . . . . . 7
⊢ (((𝐴 ∪ 𝐵) ⊆ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∧ ({𝑋} ∪ (𝐴 ∪ 𝐵)) ∈ V) → (𝐴 ∪ 𝐵) ∈ V) |
13 | 2, 11, 12 | sylancr 694 |
. . . . . 6
⊢ (𝑅 ∈ TosetRel → (𝐴 ∪ 𝐵) ∈ V) |
14 | | ssexg 4732 |
. . . . . 6
⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐴 ∈ V) |
15 | 1, 13, 14 | sylancr 694 |
. . . . 5
⊢ (𝑅 ∈ TosetRel → 𝐴 ∈ V) |
16 | | ssun2 3739 |
. . . . . 6
⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) |
17 | | ssexg 4732 |
. . . . . 6
⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ V) → 𝐵 ∈ V) |
18 | 16, 13, 17 | sylancr 694 |
. . . . 5
⊢ (𝑅 ∈ TosetRel → 𝐵 ∈ V) |
19 | | elfiun 8219 |
. . . . 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 691 |
. . . 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 20802 |
. . . . . . . 8
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐴) = 𝐴) |
22 | 21, 1 | syl6eqss 3618 |
. . . . . . 7
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐴) ⊆ (𝐴 ∪ 𝐵)) |
23 | | ssun1 3738 |
. . . . . . 7
⊢ (𝐴 ∪ 𝐵) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶) |
24 | 22, 23 | syl6ss 3580 |
. . . . . 6
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐴) ⊆
((𝐴 ∪ 𝐵) ∪ 𝐶)) |
25 | 24 | sseld 3567 |
. . . . 5
⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐴) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
26 | | cnvtsr 17045 |
. . . . . . . . . 10
⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel ) |
27 | | df-rn 5049 |
. . . . . . . . . . 11
⊢ ran 𝑅 = dom ◡𝑅 |
28 | | eqid 2610 |
. . . . . . . . . . 11
⊢ ran
(𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) |
29 | 27, 28 | ordtbaslem 20802 |
. . . . . . . . . 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 17044 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ TosetRel → 𝑅 ∈
PosetRel) |
32 | 3 | psrn 17032 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅) |
33 | 31, 32 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ TosetRel → 𝑋 = ran 𝑅) |
34 | | vex 3176 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑦 ∈ V |
35 | | vex 3176 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑥 ∈ V |
36 | 34, 35 | brcnv 5227 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦) |
37 | 36 | bicomi 213 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥𝑅𝑦 ↔ 𝑦◡𝑅𝑥) |
38 | 37 | notbii 309 |
. . . . . . . . . . . . . . 15
⊢ (¬
𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥) |
39 | 38 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ TosetRel → (¬
𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥)) |
40 | 33, 39 | rabeqbidv 3168 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ TosetRel → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}) |
41 | 33, 40 | mpteq12dv 4663 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ TosetRel → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) |
42 | 41 | rneqd 5274 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ TosetRel → ran
(𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) |
43 | 5, 42 | syl5eq 2656 |
. . . . . . . . . 10
⊢ (𝑅 ∈ TosetRel → 𝐵 = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥})) |
44 | 43 | fveq2d 6107 |
. . . . . . . . 9
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) =
(fi‘ran (𝑥 ∈ ran
𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦◡𝑅𝑥}))) |
45 | 30, 44, 43 | 3eqtr4d 2654 |
. . . . . . . 8
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) = 𝐵) |
46 | 45, 16 | syl6eqss 3618 |
. . . . . . 7
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) ⊆ (𝐴 ∪ 𝐵)) |
47 | 46, 23 | syl6ss 3580 |
. . . . . 6
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) ⊆
((𝐴 ∪ 𝐵) ∪ 𝐶)) |
48 | 47 | sseld 3567 |
. . . . 5
⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘𝐵) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
49 | | ssun2 3739 |
. . . . . . . 8
⊢ 𝐶 ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶) |
50 | 21, 4 | syl6eq 2660 |
. . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐴) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})) |
51 | 50 | eleq2d 2673 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ 𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}))) |
52 | | vex 3176 |
. . . . . . . . . . . . . . 15
⊢ 𝑚 ∈ V |
53 | | breq2 4587 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑎 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑎)) |
54 | 53 | notbid 307 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑎 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑎)) |
55 | 54 | rabbidv 3164 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) |
56 | 55 | cbvmptv 4678 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑎 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) |
57 | 56 | elrnmpt 5293 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ V → (𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})) |
58 | 52, 57 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) |
59 | 51, 58 | syl6bb 275 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ TosetRel → (𝑚 ∈ (fi‘𝐴) ↔ ∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})) |
60 | 45, 5 | syl6eq 2660 |
. . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ TosetRel →
(fi‘𝐵) = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})) |
61 | 60 | eleq2d 2673 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ 𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}))) |
62 | | vex 3176 |
. . . . . . . . . . . . . . 15
⊢ 𝑛 ∈ V |
63 | | breq1 4586 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑏 → (𝑥𝑅𝑦 ↔ 𝑏𝑅𝑦)) |
64 | 63 | notbid 307 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑏 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑏𝑅𝑦)) |
65 | 64 | rabbidv 3164 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑏 → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) |
66 | 65 | cbvmptv 4678 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) |
67 | 66 | elrnmpt 5293 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ V → (𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})) |
68 | 62, 67 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) |
69 | 61, 68 | syl6bb 275 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ TosetRel → (𝑛 ∈ (fi‘𝐵) ↔ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})) |
70 | 59, 69 | anbi12d 743 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) ↔ (∃𝑎 ∈ 𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}))) |
71 | | reeanv 3086 |
. . . . . . . . . . . . 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})) |
72 | | ineq12 3771 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚 ∩ 𝑛) = ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})) |
73 | | inrab 3858 |
. . . . . . . . . . . . . . . 16
⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} |
74 | 72, 73 | syl6eq 2660 |
. . . . . . . . . . . . . . 15
⊢ ((𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
75 | 74 | reximi 2994 |
. . . . . . . . . . . . . 14
⊢
(∃𝑏 ∈
𝑋 (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
76 | 75 | reximi 2994 |
. . . . . . . . . . . . 13
⊢
(∃𝑎 ∈
𝑋 ∃𝑏 ∈ 𝑋 (𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
77 | 71, 76 | sylbir 224 |
. . . . . . . . . . . 12
⊢
((∃𝑎 ∈
𝑋 𝑚 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∧ ∃𝑏 ∈ 𝑋 𝑛 = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
78 | 70, 77 | syl6bi 242 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ TosetRel → ((𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵)) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})) |
79 | 78 | imp 444 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → ∃𝑎 ∈ 𝑋 ∃𝑏 ∈ 𝑋 (𝑚 ∩ 𝑛) = {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
80 | 52 | inex1 4727 |
. . . . . . . . . . 11
⊢ (𝑚 ∩ 𝑛) ∈ V |
81 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) = (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
82 | 81 | elrnmpt2g 6670 |
. . . . . . . . . . 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)})) |
83 | 80, 82 | 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)}) |
84 | 79, 83 | sylibr 223 |
. . . . . . . . 9
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})) |
85 | | ordtval.4 |
. . . . . . . . 9
⊢ 𝐶 = ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) |
86 | 84, 85 | syl6eleqr 2699 |
. . . . . . . 8
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ 𝐶) |
87 | 49, 86 | sseldi 3566 |
. . . . . . 7
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑚 ∩ 𝑛) ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶)) |
88 | | eleq1 2676 |
. . . . . . 7
⊢ (𝑧 = (𝑚 ∩ 𝑛) → (𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶) ↔ (𝑚 ∩ 𝑛) ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
89 | 87, 88 | syl5ibrcom 236 |
. . . . . 6
⊢ ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘𝐴) ∧ 𝑛 ∈ (fi‘𝐵))) → (𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
90 | 89 | rexlimdvva 3020 |
. . . . 5
⊢ (𝑅 ∈ TosetRel →
(∃𝑚 ∈
(fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
91 | 25, 48, 90 | 3jaod 1384 |
. . . 4
⊢ (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘𝐴) ∨ 𝑧 ∈ (fi‘𝐵) ∨ ∃𝑚 ∈ (fi‘𝐴)∃𝑛 ∈ (fi‘𝐵)𝑧 = (𝑚 ∩ 𝑛)) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
92 | 20, 91 | sylbid 229 |
. . 3
⊢ (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴 ∪ 𝐵)) → 𝑧 ∈ ((𝐴 ∪ 𝐵) ∪ 𝐶))) |
93 | 92 | ssrdv 3574 |
. 2
⊢ (𝑅 ∈ TosetRel →
(fi‘(𝐴 ∪ 𝐵)) ⊆ ((𝐴 ∪ 𝐵) ∪ 𝐶)) |
94 | | ssfii 8208 |
. . . 4
⊢ ((𝐴 ∪ 𝐵) ∈ V → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵))) |
95 | 13, 94 | syl 17 |
. . 3
⊢ (𝑅 ∈ TosetRel → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵))) |
96 | 95 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝐴 ∪ 𝐵) ⊆ (fi‘(𝐴 ∪ 𝐵))) |
97 | | simprl 790 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑎 ∈ 𝑋) |
98 | | eqidd 2611 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) |
99 | 55 | eqeq2d 2620 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑎 → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎})) |
100 | 99 | rspcev 3282 |
. . . . . . . . . . . . . 14
⊢ ((𝑎 ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎}) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) |
101 | 97, 98, 100 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) |
102 | 8 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑋 ∈ V) |
103 | | rabexg 4739 |
. . . . . . . . . . . . . 14
⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V) |
104 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) |
105 | 104 | elrnmpt 5293 |
. . . . . . . . . . . . . 14
⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})) |
106 | 102, 103,
105 | 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})) |
107 | 101, 106 | mpbird 246 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})) |
108 | 107, 4 | syl6eleqr 2699 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ 𝐴) |
109 | 1, 108 | sseldi 3566 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (𝐴 ∪ 𝐵)) |
110 | 96, 109 | sseldd 3569 |
. . . . . . . . 9
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴 ∪ 𝐵))) |
111 | | simprr 792 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → 𝑏 ∈ 𝑋) |
112 | | eqidd 2611 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) |
113 | 65 | eqeq2d 2620 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑏 → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦})) |
114 | 113 | rspcev 3282 |
. . . . . . . . . . . . . 14
⊢ ((𝑏 ∈ 𝑋 ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) |
115 | 111, 112,
114 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) |
116 | | rabexg 4739 |
. . . . . . . . . . . . . 14
⊢ (𝑋 ∈ V → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V) |
117 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) |
118 | 117 | elrnmpt 5293 |
. . . . . . . . . . . . . 14
⊢ ({𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ↔ ∃𝑥 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} = {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})) |
119 | 102, 116,
118 | 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})) |
120 | 115, 119 | mpbird 246 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})) |
121 | 120, 5 | syl6eleqr 2699 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ 𝐵) |
122 | 16, 121 | sseldi 3566 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (𝐴 ∪ 𝐵)) |
123 | 96, 122 | sseldd 3569 |
. . . . . . . . 9
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴 ∪ 𝐵))) |
124 | | fiin 8211 |
. . . . . . . . 9
⊢ (({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∈ (fi‘(𝐴 ∪ 𝐵)) ∧ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦} ∈ (fi‘(𝐴 ∪ 𝐵))) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴 ∪ 𝐵))) |
125 | 110, 123,
124 | syl2anc 691 |
. . . . . . . 8
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑎} ∩ {𝑦 ∈ 𝑋 ∣ ¬ 𝑏𝑅𝑦}) ∈ (fi‘(𝐴 ∪ 𝐵))) |
126 | 73, 125 | syl5eqelr 2693 |
. . . . . . 7
⊢ ((𝑅 ∈ TosetRel ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴 ∪ 𝐵))) |
127 | 126 | ralrimivva 2954 |
. . . . . 6
⊢ (𝑅 ∈ TosetRel →
∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)} ∈ (fi‘(𝐴 ∪ 𝐵))) |
128 | 81 | fmpt2 7126 |
. . . . . 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))) |
129 | 127, 128 | sylib 207 |
. . . . 5
⊢ (𝑅 ∈ TosetRel → (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴 ∪ 𝐵))) |
130 | | frn 5966 |
. . . . 5
⊢ ((𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}):(𝑋 × 𝑋)⟶(fi‘(𝐴 ∪ 𝐵)) → ran (𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ⊆ (fi‘(𝐴 ∪ 𝐵))) |
131 | 129, 130 | syl 17 |
. . . 4
⊢ (𝑅 ∈ TosetRel → ran
(𝑎 ∈ 𝑋, 𝑏 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)}) ⊆ (fi‘(𝐴 ∪ 𝐵))) |
132 | 85, 131 | syl5eqss 3612 |
. . 3
⊢ (𝑅 ∈ TosetRel → 𝐶 ⊆ (fi‘(𝐴 ∪ 𝐵))) |
133 | 95, 132 | unssd 3751 |
. 2
⊢ (𝑅 ∈ TosetRel → ((𝐴 ∪ 𝐵) ∪ 𝐶) ⊆ (fi‘(𝐴 ∪ 𝐵))) |
134 | 93, 133 | eqssd 3585 |
1
⊢ (𝑅 ∈ TosetRel →
(fi‘(𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∪ 𝐶)) |