Step | Hyp | Ref
| Expression |
1 | | csbeq1 3924 |
. . . . . . . 8
⊢ (𝑠 = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) → ⦋𝑠 / 𝑥⦌𝑆 = ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆) |
2 | | csbeq1 3924 |
. . . . . . . 8
⊢ (𝑠 = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) → ⦋𝑠 / 𝑥⦌𝐵 = ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) |
3 | 1, 2 | freq12d 5669 |
. . . . . . 7
⊢ (𝑠 = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) → (⦋𝑠 / 𝑥⦌𝑆 Fr ⦋𝑠 / 𝑥⦌𝐵 ↔ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆 Fr ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵)) |
4 | | simpl3 1193 |
. . . . . . . 8
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) |
5 | | nfv 1913 |
. . . . . . . . 9
⊢
Ⅎ𝑠 𝑆 Fr 𝐵 |
6 | | nfcsb1v 3946 |
. . . . . . . . . 10
⊢
Ⅎ𝑥⦋𝑠 / 𝑥⦌𝑆 |
7 | | nfcsb1v 3946 |
. . . . . . . . . 10
⊢
Ⅎ𝑥⦋𝑠 / 𝑥⦌𝐵 |
8 | 6, 7 | nffr 5673 |
. . . . . . . . 9
⊢
Ⅎ𝑥⦋𝑠 / 𝑥⦌𝑆 Fr ⦋𝑠 / 𝑥⦌𝐵 |
9 | | csbeq1a 3935 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑠 → 𝑆 = ⦋𝑠 / 𝑥⦌𝑆) |
10 | | csbeq1a 3935 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑠 → 𝐵 = ⦋𝑠 / 𝑥⦌𝐵) |
11 | 9, 10 | freq12d 5669 |
. . . . . . . . 9
⊢ (𝑥 = 𝑠 → (𝑆 Fr 𝐵 ↔ ⦋𝑠 / 𝑥⦌𝑆 Fr ⦋𝑠 / 𝑥⦌𝐵)) |
12 | 5, 8, 11 | cbvralw 3312 |
. . . . . . . 8
⊢
(∀𝑥 ∈
𝐴 𝑆 Fr 𝐵 ↔ ∀𝑠 ∈ 𝐴 ⦋𝑠 / 𝑥⦌𝑆 Fr ⦋𝑠 / 𝑥⦌𝐵) |
13 | 4, 12 | sylib 218 |
. . . . . . 7
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → ∀𝑠 ∈ 𝐴 ⦋𝑠 / 𝑥⦌𝑆 Fr ⦋𝑠 / 𝑥⦌𝐵) |
14 | | weiun.1 |
. . . . . . . . . . 11
⊢ 𝐹 = (𝑤 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢)) |
15 | | weiun.2 |
. . . . . . . . . . 11
⊢ 𝑇 = {〈𝑦, 𝑧〉 ∣ ((𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))} |
16 | | simpl1 1191 |
. . . . . . . . . . 11
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → 𝑅 We 𝐴) |
17 | | simpl2 1192 |
. . . . . . . . . . 11
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → 𝑅 Se 𝐴) |
18 | 14, 15, 16, 17 | weiunlem2 36429 |
. . . . . . . . . 10
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → (𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑡 ∈ ∪
𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵 ∧ ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ ⦋ 𝑠 / 𝑥⦌𝐵 ¬ 𝑠𝑅(𝐹‘𝑡))) |
19 | 18 | simp1d 1142 |
. . . . . . . . 9
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴) |
20 | 19 | fimassd 6768 |
. . . . . . . 8
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → (𝐹 “ 𝑟) ⊆ 𝐴) |
21 | | eqid 2740 |
. . . . . . . . . 10
⊢
(℩𝑝
∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) |
22 | | simprl 770 |
. . . . . . . . . 10
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → 𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵) |
23 | | simprr 772 |
. . . . . . . . . 10
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → 𝑟 ≠ ∅) |
24 | 14, 15, 16, 17, 21, 22, 23 | weiunfrlem 36430 |
. . . . . . . . 9
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → ((℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) ∈ (𝐹 “ 𝑟) ∧ ∀𝑡 ∈ 𝑟 ¬ (𝐹‘𝑡)𝑅(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) ∧ ∀𝑡 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵)(𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) |
25 | 24 | simp1d 1142 |
. . . . . . . 8
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) ∈ (𝐹 “ 𝑟)) |
26 | 20, 25 | sseldd 4009 |
. . . . . . 7
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) ∈ 𝐴) |
27 | 3, 13, 26 | rspcdva 3636 |
. . . . . 6
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) →
⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆 Fr ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) |
28 | | inss2 4259 |
. . . . . . 7
⊢ (𝑟 ∩
⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ⊆
⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵 |
29 | 28 | a1i 11 |
. . . . . 6
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ⊆
⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) |
30 | | vex 3492 |
. . . . . . . 8
⊢ 𝑟 ∈ V |
31 | 30 | inex1 5335 |
. . . . . . 7
⊢ (𝑟 ∩
⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∈ V |
32 | 31 | a1i 11 |
. . . . . 6
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∈ V) |
33 | 19 | ffund 6751 |
. . . . . . . 8
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → Fun 𝐹) |
34 | | fvelima 6987 |
. . . . . . . 8
⊢ ((Fun
𝐹 ∧
(℩𝑝 ∈
(𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) ∈ (𝐹 “ 𝑟)) → ∃𝑡 ∈ 𝑟 (𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)) |
35 | 33, 25, 34 | syl2anc 583 |
. . . . . . 7
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → ∃𝑡 ∈ 𝑟 (𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)) |
36 | | simprl 770 |
. . . . . . . . 9
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑡 ∈ 𝑟 ∧ (𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) → 𝑡 ∈ 𝑟) |
37 | | simplrl 776 |
. . . . . . . . . . . 12
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑡 ∈ 𝑟 ∧ (𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) → 𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵) |
38 | 37, 36 | sseldd 4009 |
. . . . . . . . . . 11
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑡 ∈ 𝑟 ∧ (𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) → 𝑡 ∈ ∪
𝑥 ∈ 𝐴 𝐵) |
39 | 18 | simp2d 1143 |
. . . . . . . . . . . 12
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵) |
40 | 39 | r19.21bi 3257 |
. . . . . . . . . . 11
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ 𝑡 ∈ ∪
𝑥 ∈ 𝐴 𝐵) → 𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵) |
41 | 38, 40 | syldan 590 |
. . . . . . . . . 10
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑡 ∈ 𝑟 ∧ (𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) → 𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵) |
42 | | simprr 772 |
. . . . . . . . . . 11
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑡 ∈ 𝑟 ∧ (𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) → (𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)) |
43 | 42 | csbeq1d 3925 |
. . . . . . . . . 10
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑡 ∈ 𝑟 ∧ (𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) → ⦋(𝐹‘𝑡) / 𝑥⦌𝐵 = ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) |
44 | 41, 43 | eleqtrd 2846 |
. . . . . . . . 9
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑡 ∈ 𝑟 ∧ (𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) → 𝑡 ∈ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) |
45 | 36, 44 | elind 4223 |
. . . . . . . 8
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑡 ∈ 𝑟 ∧ (𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) → 𝑡 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵)) |
46 | 45 | ne0d 4365 |
. . . . . . 7
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑡 ∈ 𝑟 ∧ (𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) → (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ≠ ∅) |
47 | 35, 46 | rexlimddv 3167 |
. . . . . 6
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ≠ ∅) |
48 | 27, 29, 32, 47 | frd 5656 |
. . . . 5
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → ∃𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵)∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛) |
49 | | simprl 770 |
. . . . . 6
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) → 𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵)) |
50 | 49 | elin1d 4227 |
. . . . 5
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) → 𝑛 ∈ 𝑟) |
51 | | fveq2 6920 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑜 → (𝐹‘𝑡) = (𝐹‘𝑜)) |
52 | 51 | breq1d 5176 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑜 → ((𝐹‘𝑡)𝑅(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) ↔ (𝐹‘𝑜)𝑅(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) |
53 | 52 | notbid 318 |
. . . . . . . . . . 11
⊢ (𝑡 = 𝑜 → (¬ (𝐹‘𝑡)𝑅(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) ↔ ¬ (𝐹‘𝑜)𝑅(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) |
54 | 24 | ad2antrr 725 |
. . . . . . . . . . . 12
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → ((℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) ∈ (𝐹 “ 𝑟) ∧ ∀𝑡 ∈ 𝑟 ¬ (𝐹‘𝑡)𝑅(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) ∧ ∀𝑡 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵)(𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) |
55 | 54 | simp2d 1143 |
. . . . . . . . . . 11
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → ∀𝑡 ∈ 𝑟 ¬ (𝐹‘𝑡)𝑅(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)) |
56 | | simpr 484 |
. . . . . . . . . . 11
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → 𝑜 ∈ 𝑟) |
57 | 53, 55, 56 | rspcdva 3636 |
. . . . . . . . . 10
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → ¬ (𝐹‘𝑜)𝑅(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)) |
58 | | fveqeq2 6929 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑛 → ((𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) ↔ (𝐹‘𝑛) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) |
59 | 54 | simp3d 1144 |
. . . . . . . . . . . 12
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → ∀𝑡 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵)(𝐹‘𝑡) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)) |
60 | | simplrl 776 |
. . . . . . . . . . . 12
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → 𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵)) |
61 | 58, 59, 60 | rspcdva 3636 |
. . . . . . . . . . 11
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → (𝐹‘𝑛) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)) |
62 | 61 | breq2d 5178 |
. . . . . . . . . 10
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → ((𝐹‘𝑜)𝑅(𝐹‘𝑛) ↔ (𝐹‘𝑜)𝑅(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝))) |
63 | 57, 62 | mtbird 325 |
. . . . . . . . 9
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → ¬ (𝐹‘𝑜)𝑅(𝐹‘𝑛)) |
64 | | breq1 5169 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑜 → (𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛 ↔ 𝑜⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) |
65 | 64 | notbid 318 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑜 → (¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛 ↔ ¬ 𝑜⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) |
66 | | simprr 772 |
. . . . . . . . . . . . . 14
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) → ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛) |
67 | 66 | ad2antrr 725 |
. . . . . . . . . . . . 13
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛) |
68 | | simplr 768 |
. . . . . . . . . . . . . 14
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → 𝑜 ∈ 𝑟) |
69 | | id 22 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑜 → 𝑡 = 𝑜) |
70 | 51 | csbeq1d 3925 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑜 → ⦋(𝐹‘𝑡) / 𝑥⦌𝐵 = ⦋(𝐹‘𝑜) / 𝑥⦌𝐵) |
71 | 69, 70 | eleq12d 2838 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑜 → (𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵 ↔ 𝑜 ∈ ⦋(𝐹‘𝑜) / 𝑥⦌𝐵)) |
72 | 39 | ad3antrrr 729 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → ∀𝑡 ∈ ∪
𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵) |
73 | 22 | ad2antrr 725 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → 𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵) |
74 | 73, 56 | sseldd 4009 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → 𝑜 ∈ ∪
𝑥 ∈ 𝐴 𝐵) |
75 | 74 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → 𝑜 ∈ ∪
𝑥 ∈ 𝐴 𝐵) |
76 | 71, 72, 75 | rspcdva 3636 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → 𝑜 ∈ ⦋(𝐹‘𝑜) / 𝑥⦌𝐵) |
77 | | simpr 484 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → (𝐹‘𝑜) = (𝐹‘𝑛)) |
78 | 61 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → (𝐹‘𝑛) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)) |
79 | 77, 78 | eqtrd 2780 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → (𝐹‘𝑜) = (℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝)) |
80 | 79 | csbeq1d 3925 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → ⦋(𝐹‘𝑜) / 𝑥⦌𝐵 = ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) |
81 | 76, 80 | eleqtrd 2846 |
. . . . . . . . . . . . . 14
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → 𝑜 ∈ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) |
82 | 68, 81 | elind 4223 |
. . . . . . . . . . . . 13
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → 𝑜 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵)) |
83 | 65, 67, 82 | rspcdva 3636 |
. . . . . . . . . . . 12
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → ¬ 𝑜⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛) |
84 | 79 | csbeq1d 3925 |
. . . . . . . . . . . . 13
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → ⦋(𝐹‘𝑜) / 𝑥⦌𝑆 = ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆) |
85 | 84 | breqd 5177 |
. . . . . . . . . . . 12
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → (𝑜⦋(𝐹‘𝑜) / 𝑥⦌𝑆𝑛 ↔ 𝑜⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) |
86 | 83, 85 | mtbird 325 |
. . . . . . . . . . 11
⊢
((((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) ∧ (𝐹‘𝑜) = (𝐹‘𝑛)) → ¬ 𝑜⦋(𝐹‘𝑜) / 𝑥⦌𝑆𝑛) |
87 | 86 | ex 412 |
. . . . . . . . . 10
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → ((𝐹‘𝑜) = (𝐹‘𝑛) → ¬ 𝑜⦋(𝐹‘𝑜) / 𝑥⦌𝑆𝑛)) |
88 | | imnan 399 |
. . . . . . . . . 10
⊢ (((𝐹‘𝑜) = (𝐹‘𝑛) → ¬ 𝑜⦋(𝐹‘𝑜) / 𝑥⦌𝑆𝑛) ↔ ¬ ((𝐹‘𝑜) = (𝐹‘𝑛) ∧ 𝑜⦋(𝐹‘𝑜) / 𝑥⦌𝑆𝑛)) |
89 | 87, 88 | sylib 218 |
. . . . . . . . 9
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → ¬ ((𝐹‘𝑜) = (𝐹‘𝑛) ∧ 𝑜⦋(𝐹‘𝑜) / 𝑥⦌𝑆𝑛)) |
90 | | pm4.56 989 |
. . . . . . . . . 10
⊢ ((¬
(𝐹‘𝑜)𝑅(𝐹‘𝑛) ∧ ¬ ((𝐹‘𝑜) = (𝐹‘𝑛) ∧ 𝑜⦋(𝐹‘𝑜) / 𝑥⦌𝑆𝑛)) ↔ ¬ ((𝐹‘𝑜)𝑅(𝐹‘𝑛) ∨ ((𝐹‘𝑜) = (𝐹‘𝑛) ∧ 𝑜⦋(𝐹‘𝑜) / 𝑥⦌𝑆𝑛))) |
91 | 90 | biimpi 216 |
. . . . . . . . 9
⊢ ((¬
(𝐹‘𝑜)𝑅(𝐹‘𝑛) ∧ ¬ ((𝐹‘𝑜) = (𝐹‘𝑛) ∧ 𝑜⦋(𝐹‘𝑜) / 𝑥⦌𝑆𝑛)) → ¬ ((𝐹‘𝑜)𝑅(𝐹‘𝑛) ∨ ((𝐹‘𝑜) = (𝐹‘𝑛) ∧ 𝑜⦋(𝐹‘𝑜) / 𝑥⦌𝑆𝑛))) |
92 | 63, 89, 91 | syl2anc 583 |
. . . . . . . 8
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → ¬ ((𝐹‘𝑜)𝑅(𝐹‘𝑛) ∨ ((𝐹‘𝑜) = (𝐹‘𝑛) ∧ 𝑜⦋(𝐹‘𝑜) / 𝑥⦌𝑆𝑛))) |
93 | 92 | intnand 488 |
. . . . . . 7
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → ¬ ((𝑜 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑛 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑜)𝑅(𝐹‘𝑛) ∨ ((𝐹‘𝑜) = (𝐹‘𝑛) ∧ 𝑜⦋(𝐹‘𝑜) / 𝑥⦌𝑆𝑛)))) |
94 | 14, 15 | weiunlem1 36428 |
. . . . . . 7
⊢ (𝑜𝑇𝑛 ↔ ((𝑜 ∈ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑛 ∈ ∪
𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑜)𝑅(𝐹‘𝑛) ∨ ((𝐹‘𝑜) = (𝐹‘𝑛) ∧ 𝑜⦋(𝐹‘𝑜) / 𝑥⦌𝑆𝑛)))) |
95 | 93, 94 | sylnibr 329 |
. . . . . 6
⊢
(((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) ∧ 𝑜 ∈ 𝑟) → ¬ 𝑜𝑇𝑛) |
96 | 95 | ralrimiva 3152 |
. . . . 5
⊢ ((((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) ∧ (𝑛 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ∧ ∀𝑚 ∈ (𝑟 ∩ ⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝐵) ¬ 𝑚⦋(℩𝑝 ∈ (𝐹 “ 𝑟)∀𝑞 ∈ (𝐹 “ 𝑟) ¬ 𝑞𝑅𝑝) / 𝑥⦌𝑆𝑛)) → ∀𝑜 ∈ 𝑟 ¬ 𝑜𝑇𝑛) |
97 | 48, 50, 96 | reximssdv 3179 |
. . . 4
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) ∧ (𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → ∃𝑛 ∈ 𝑟 ∀𝑜 ∈ 𝑟 ¬ 𝑜𝑇𝑛) |
98 | 97 | ex 412 |
. . 3
⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) → ((𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅) → ∃𝑛 ∈ 𝑟 ∀𝑜 ∈ 𝑟 ¬ 𝑜𝑇𝑛)) |
99 | 98 | alrimiv 1926 |
. 2
⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) → ∀𝑟((𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅) → ∃𝑛 ∈ 𝑟 ∀𝑜 ∈ 𝑟 ¬ 𝑜𝑇𝑛)) |
100 | | df-fr 5652 |
. 2
⊢ (𝑇 Fr ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑟((𝑟 ⊆ ∪
𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅) → ∃𝑛 ∈ 𝑟 ∀𝑜 ∈ 𝑟 ¬ 𝑜𝑇𝑛)) |
101 | 99, 100 | sylibr 234 |
1
⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) → 𝑇 Fr ∪
𝑥 ∈ 𝐴 𝐵) |