Step |
Hyp |
Ref |
Expression |
1 |
|
weiun.1 |
⊢ 𝐹 = ( 𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ ( ℩ 𝑢 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑢 ) ) |
2 |
|
weiun.2 |
⊢ 𝑇 = { 〈 𝑦 , 𝑧 〉 ∣ ( ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑦 ) 𝑅 ( 𝐹 ‘ 𝑧 ) ∨ ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ∧ 𝑦 ⦋ ( 𝐹 ‘ 𝑦 ) / 𝑥 ⦌ 𝑆 𝑧 ) ) ) } |
3 |
|
csbeq1 |
⊢ ( 𝑠 = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) → ⦋ 𝑠 / 𝑥 ⦌ 𝑆 = ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 ) |
4 |
|
csbeq1 |
⊢ ( 𝑠 = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) → ⦋ 𝑠 / 𝑥 ⦌ 𝐵 = ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
5 |
3 4
|
freq12d |
⊢ ( 𝑠 = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) → ( ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Fr ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ↔ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 Fr ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ) |
6 |
|
simpl3 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) |
7 |
|
nfv |
⊢ Ⅎ 𝑠 𝑆 Fr 𝐵 |
8 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑠 / 𝑥 ⦌ 𝑆 |
9 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑠 / 𝑥 ⦌ 𝐵 |
10 |
8 9
|
nffr |
⊢ Ⅎ 𝑥 ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Fr ⦋ 𝑠 / 𝑥 ⦌ 𝐵 |
11 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑠 → 𝑆 = ⦋ 𝑠 / 𝑥 ⦌ 𝑆 ) |
12 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑠 → 𝐵 = ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) |
13 |
11 12
|
freq12d |
⊢ ( 𝑥 = 𝑠 → ( 𝑆 Fr 𝐵 ↔ ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Fr ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) |
14 |
7 10 13
|
cbvralw |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ↔ ∀ 𝑠 ∈ 𝐴 ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Fr ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) |
15 |
6 14
|
sylib |
⊢ ( ( ( 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 ) |
16 |
|
simpl1 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → 𝑅 We 𝐴 ) |
17 |
|
simpl2 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → 𝑅 Se 𝐴 ) |
18 |
1 2 16 17
|
weiunlem2 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → ( 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ∧ ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) ) |
19 |
18
|
simp1d |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ) |
20 |
19
|
fimassd |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → ( 𝐹 “ 𝑟 ) ⊆ 𝐴 ) |
21 |
|
eqid |
⊢ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) |
22 |
|
simprl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
23 |
|
simprr |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → 𝑟 ≠ ∅ ) |
24 |
1 2 16 17 21 22 23
|
weiunfrlem |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → ( ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ∈ ( 𝐹 “ 𝑟 ) ∧ ∀ 𝑡 ∈ 𝑟 ¬ ( 𝐹 ‘ 𝑡 ) 𝑅 ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ∧ ∀ 𝑡 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ( 𝐹 ‘ 𝑡 ) = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) ) |
25 |
24
|
simp1d |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ∈ ( 𝐹 “ 𝑟 ) ) |
26 |
20 25
|
sseldd |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ∈ 𝐴 ) |
27 |
5 15 26
|
rspcdva |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 Fr ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
28 |
|
inss2 |
⊢ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ⊆ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 |
29 |
28
|
a1i |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ⊆ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
30 |
|
vex |
⊢ 𝑟 ∈ V |
31 |
30
|
inex1 |
⊢ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∈ V |
32 |
31
|
a1i |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∈ V ) |
33 |
19
|
ffund |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → Fun 𝐹 ) |
34 |
|
fvelima |
⊢ ( ( Fun 𝐹 ∧ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ∈ ( 𝐹 “ 𝑟 ) ) → ∃ 𝑡 ∈ 𝑟 ( 𝐹 ‘ 𝑡 ) = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) |
35 |
33 25 34
|
syl2anc |
⊢ ( ( ( 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 ) ¬ 𝑞 𝑅 𝑝 ) ) |
36 |
|
simprl |
⊢ ( ( ( ( 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 ) ¬ 𝑞 𝑅 𝑝 ) ) ) → 𝑡 ∈ 𝑟 ) |
37 |
|
simplrl |
⊢ ( ( ( ( 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 ) ¬ 𝑞 𝑅 𝑝 ) ) ) → 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
38 |
37 36
|
sseldd |
⊢ ( ( ( ( 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 ) ¬ 𝑞 𝑅 𝑝 ) ) ) → 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
39 |
18
|
simp2d |
⊢ ( ( ( 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 ) |
40 |
39
|
r19.21bi |
⊢ ( ( ( ( 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 |
⊢ ( ( ( ( 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 ) ¬ 𝑞 𝑅 𝑝 ) ) ) → 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ) |
42 |
|
simprr |
⊢ ( ( ( ( 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 ) ¬ 𝑞 𝑅 𝑝 ) ) ) → ( 𝐹 ‘ 𝑡 ) = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) |
43 |
42
|
csbeq1d |
⊢ ( ( ( ( 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 ) ¬ 𝑞 𝑅 𝑝 ) ) ) → ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 = ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
44 |
41 43
|
eleqtrd |
⊢ ( ( ( ( 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 ) ¬ 𝑞 𝑅 𝑝 ) ) ) → 𝑡 ∈ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
45 |
36 44
|
elind |
⊢ ( ( ( ( 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 ) ¬ 𝑞 𝑅 𝑝 ) ) ) → 𝑡 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ) |
46 |
45
|
ne0d |
⊢ ( ( ( ( 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 ) ¬ 𝑞 𝑅 𝑝 ) ) ) → ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) |
47 |
35 46
|
rexlimddv |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) |
48 |
27 29 32 47
|
frd |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → ∃ 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) |
49 |
|
simprl |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) → 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ) |
50 |
49
|
elin1d |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) → 𝑛 ∈ 𝑟 ) |
51 |
|
fveq2 |
⊢ ( 𝑡 = 𝑜 → ( 𝐹 ‘ 𝑡 ) = ( 𝐹 ‘ 𝑜 ) ) |
52 |
51
|
breq1d |
⊢ ( 𝑡 = 𝑜 → ( ( 𝐹 ‘ 𝑡 ) 𝑅 ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ↔ ( 𝐹 ‘ 𝑜 ) 𝑅 ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) ) |
53 |
52
|
notbid |
⊢ ( 𝑡 = 𝑜 → ( ¬ ( 𝐹 ‘ 𝑡 ) 𝑅 ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ↔ ¬ ( 𝐹 ‘ 𝑜 ) 𝑅 ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) ) |
54 |
24
|
ad2antrr |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → ( ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ∈ ( 𝐹 “ 𝑟 ) ∧ ∀ 𝑡 ∈ 𝑟 ¬ ( 𝐹 ‘ 𝑡 ) 𝑅 ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ∧ ∀ 𝑡 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ( 𝐹 ‘ 𝑡 ) = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) ) |
55 |
54
|
simp2d |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → ∀ 𝑡 ∈ 𝑟 ¬ ( 𝐹 ‘ 𝑡 ) 𝑅 ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) |
56 |
|
simpr |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → 𝑜 ∈ 𝑟 ) |
57 |
53 55 56
|
rspcdva |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → ¬ ( 𝐹 ‘ 𝑜 ) 𝑅 ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) |
58 |
|
fveqeq2 |
⊢ ( 𝑡 = 𝑛 → ( ( 𝐹 ‘ 𝑡 ) = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ↔ ( 𝐹 ‘ 𝑛 ) = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) ) |
59 |
54
|
simp3d |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → ∀ 𝑡 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ( 𝐹 ‘ 𝑡 ) = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) |
60 |
|
simplrl |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ) |
61 |
58 59 60
|
rspcdva |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → ( 𝐹 ‘ 𝑛 ) = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) |
62 |
61
|
breq2d |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → ( ( 𝐹 ‘ 𝑜 ) 𝑅 ( 𝐹 ‘ 𝑛 ) ↔ ( 𝐹 ‘ 𝑜 ) 𝑅 ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) ) |
63 |
57 62
|
mtbird |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → ¬ ( 𝐹 ‘ 𝑜 ) 𝑅 ( 𝐹 ‘ 𝑛 ) ) |
64 |
|
breq1 |
⊢ ( 𝑚 = 𝑜 → ( 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ↔ 𝑜 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) |
65 |
64
|
notbid |
⊢ ( 𝑚 = 𝑜 → ( ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ↔ ¬ 𝑜 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) |
66 |
|
simprr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) → ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) |
67 |
66
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) |
68 |
|
simplr |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → 𝑜 ∈ 𝑟 ) |
69 |
|
id |
⊢ ( 𝑡 = 𝑜 → 𝑡 = 𝑜 ) |
70 |
51
|
csbeq1d |
⊢ ( 𝑡 = 𝑜 → ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 = ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝐵 ) |
71 |
69 70
|
eleq12d |
⊢ ( 𝑡 = 𝑜 → ( 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ↔ 𝑜 ∈ ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝐵 ) ) |
72 |
39
|
ad3antrrr |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ) |
73 |
22
|
ad2antrr |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
74 |
73 56
|
sseldd |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → 𝑜 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
75 |
74
|
adantr |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → 𝑜 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
76 |
71 72 75
|
rspcdva |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → 𝑜 ∈ ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝐵 ) |
77 |
|
simpr |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) |
78 |
61
|
adantr |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑛 ) = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) |
79 |
77 78
|
eqtrd |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → ( 𝐹 ‘ 𝑜 ) = ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) ) |
80 |
79
|
csbeq1d |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝐵 = ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
81 |
76 80
|
eleqtrd |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → 𝑜 ∈ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
82 |
68 81
|
elind |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → 𝑜 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ) |
83 |
65 67 82
|
rspcdva |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → ¬ 𝑜 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) |
84 |
79
|
csbeq1d |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 = ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 ) |
85 |
84
|
breqd |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → ( 𝑜 ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 𝑛 ↔ 𝑜 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) |
86 |
83 85
|
mtbird |
⊢ ( ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) ∧ ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ) → ¬ 𝑜 ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 𝑛 ) |
87 |
86
|
ex |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → ( ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) → ¬ 𝑜 ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) |
88 |
|
imnan |
⊢ ( ( ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) → ¬ 𝑜 ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 𝑛 ) ↔ ¬ ( ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ∧ 𝑜 ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) |
89 |
87 88
|
sylib |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → ¬ ( ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ∧ 𝑜 ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) |
90 |
|
pm4.56 |
⊢ ( ( ¬ ( 𝐹 ‘ 𝑜 ) 𝑅 ( 𝐹 ‘ 𝑛 ) ∧ ¬ ( ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ∧ 𝑜 ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ↔ ¬ ( ( 𝐹 ‘ 𝑜 ) 𝑅 ( 𝐹 ‘ 𝑛 ) ∨ ( ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ∧ 𝑜 ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ) |
91 |
90
|
biimpi |
⊢ ( ( ¬ ( 𝐹 ‘ 𝑜 ) 𝑅 ( 𝐹 ‘ 𝑛 ) ∧ ¬ ( ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ∧ 𝑜 ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) → ¬ ( ( 𝐹 ‘ 𝑜 ) 𝑅 ( 𝐹 ‘ 𝑛 ) ∨ ( ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ∧ 𝑜 ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ) |
92 |
63 89 91
|
syl2anc |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → ¬ ( ( 𝐹 ‘ 𝑜 ) 𝑅 ( 𝐹 ‘ 𝑛 ) ∨ ( ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ∧ 𝑜 ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ) |
93 |
92
|
intnand |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → ¬ ( ( 𝑜 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑛 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑜 ) 𝑅 ( 𝐹 ‘ 𝑛 ) ∨ ( ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ∧ 𝑜 ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ) ) |
94 |
1 2
|
weiunlem1 |
⊢ ( 𝑜 𝑇 𝑛 ↔ ( ( 𝑜 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑛 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑜 ) 𝑅 ( 𝐹 ‘ 𝑛 ) ∨ ( ( 𝐹 ‘ 𝑜 ) = ( 𝐹 ‘ 𝑛 ) ∧ 𝑜 ⦋ ( 𝐹 ‘ 𝑜 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ) ) |
95 |
93 94
|
sylnibr |
⊢ ( ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) ∧ 𝑜 ∈ 𝑟 ) → ¬ 𝑜 𝑇 𝑛 ) |
96 |
95
|
ralrimiva |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) ∧ ( 𝑛 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ∧ ∀ 𝑚 ∈ ( 𝑟 ∩ ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝐵 ) ¬ 𝑚 ⦋ ( ℩ 𝑝 ∈ ( 𝐹 “ 𝑟 ) ∀ 𝑞 ∈ ( 𝐹 “ 𝑟 ) ¬ 𝑞 𝑅 𝑝 ) / 𝑥 ⦌ 𝑆 𝑛 ) ) → ∀ 𝑜 ∈ 𝑟 ¬ 𝑜 𝑇 𝑛 ) |
97 |
48 50 96
|
reximssdv |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) ∧ ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) ) → ∃ 𝑛 ∈ 𝑟 ∀ 𝑜 ∈ 𝑟 ¬ 𝑜 𝑇 𝑛 ) |
98 |
97
|
ex |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) → ( ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) → ∃ 𝑛 ∈ 𝑟 ∀ 𝑜 ∈ 𝑟 ¬ 𝑜 𝑇 𝑛 ) ) |
99 |
98
|
alrimiv |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) → ∀ 𝑟 ( ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) → ∃ 𝑛 ∈ 𝑟 ∀ 𝑜 ∈ 𝑟 ¬ 𝑜 𝑇 𝑛 ) ) |
100 |
|
df-fr |
⊢ ( 𝑇 Fr ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑟 ( ( 𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅ ) → ∃ 𝑛 ∈ 𝑟 ∀ 𝑜 ∈ 𝑟 ¬ 𝑜 𝑇 𝑛 ) ) |
101 |
99 100
|
sylibr |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Fr 𝐵 ) → 𝑇 Fr ∪ 𝑥 ∈ 𝐴 𝐵 ) |