| 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 |
|
simpl1 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑅 We 𝐴 ) |
| 4 |
|
weso |
⊢ ( 𝑅 We 𝐴 → 𝑅 Or 𝐴 ) |
| 5 |
3 4
|
syl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑅 Or 𝐴 ) |
| 6 |
|
simpl2 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑅 Se 𝐴 ) |
| 7 |
1 2 3 6
|
weiunlem2 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ∧ ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) ) |
| 8 |
7
|
simp1d |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ) |
| 9 |
|
simpr1 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
| 10 |
8 9
|
ffvelcdmd |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝐹 ‘ 𝑝 ) ∈ 𝐴 ) |
| 11 |
|
sonr |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐹 ‘ 𝑝 ) ∈ 𝐴 ) → ¬ ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑝 ) ) |
| 12 |
5 10 11
|
syl2anc |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ¬ ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑝 ) ) |
| 13 |
|
csbeq1 |
⊢ ( 𝑠 = ( 𝐹 ‘ 𝑝 ) → ⦋ 𝑠 / 𝑥 ⦌ 𝑆 = ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 ) |
| 14 |
|
csbeq1 |
⊢ ( 𝑠 = ( 𝐹 ‘ 𝑝 ) → ⦋ 𝑠 / 𝑥 ⦌ 𝐵 = ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
| 15 |
13 14
|
poeq12d |
⊢ ( 𝑠 = ( 𝐹 ‘ 𝑝 ) → ( ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Po ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ↔ ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 Po ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ) ) |
| 16 |
|
simpl3 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) |
| 17 |
|
nfv |
⊢ Ⅎ 𝑠 𝑆 Po 𝐵 |
| 18 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑠 / 𝑥 ⦌ 𝑆 |
| 19 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑠 / 𝑥 ⦌ 𝐵 |
| 20 |
18 19
|
nfpo |
⊢ Ⅎ 𝑥 ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Po ⦋ 𝑠 / 𝑥 ⦌ 𝐵 |
| 21 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑠 → 𝑆 = ⦋ 𝑠 / 𝑥 ⦌ 𝑆 ) |
| 22 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑠 → 𝐵 = ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) |
| 23 |
21 22
|
poeq12d |
⊢ ( 𝑥 = 𝑠 → ( 𝑆 Po 𝐵 ↔ ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Po ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) |
| 24 |
17 20 23
|
cbvralw |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ↔ ∀ 𝑠 ∈ 𝐴 ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Po ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) |
| 25 |
16 24
|
sylib |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ∀ 𝑠 ∈ 𝐴 ⦋ 𝑠 / 𝑥 ⦌ 𝑆 Po ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) |
| 26 |
15 25 10
|
rspcdva |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 Po ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
| 27 |
|
id |
⊢ ( 𝑡 = 𝑝 → 𝑡 = 𝑝 ) |
| 28 |
|
fveq2 |
⊢ ( 𝑡 = 𝑝 → ( 𝐹 ‘ 𝑡 ) = ( 𝐹 ‘ 𝑝 ) ) |
| 29 |
28
|
csbeq1d |
⊢ ( 𝑡 = 𝑝 → ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 = ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
| 30 |
27 29
|
eleq12d |
⊢ ( 𝑡 = 𝑝 → ( 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ↔ 𝑝 ∈ ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ) ) |
| 31 |
7
|
simp2d |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ) |
| 32 |
30 31 9
|
rspcdva |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑝 ∈ ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
| 33 |
|
poirr |
⊢ ( ( ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 Po ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ∧ 𝑝 ∈ ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ) → ¬ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑝 ) |
| 34 |
26 32 33
|
syl2anc |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ¬ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑝 ) |
| 35 |
34
|
intnand |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ¬ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑝 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑝 ) ) |
| 36 |
|
ioran |
⊢ ( ¬ ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑝 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑝 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑝 ) ) ↔ ( ¬ ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑝 ) ∧ ¬ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑝 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑝 ) ) ) |
| 37 |
12 35 36
|
sylanbrc |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ¬ ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑝 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑝 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑝 ) ) ) |
| 38 |
1 2
|
weiunlem1 |
⊢ ( 𝑝 𝑇 𝑝 ↔ ( ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑝 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑝 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑝 ) ) ) ) |
| 39 |
38
|
simprbi |
⊢ ( 𝑝 𝑇 𝑝 → ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑝 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑝 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑝 ) ) ) |
| 40 |
37 39
|
nsyl |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ¬ 𝑝 𝑇 𝑝 ) |
| 41 |
|
simpr3 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
| 42 |
9 41
|
jca |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) |
| 43 |
1 2
|
weiunlem1 |
⊢ ( 𝑝 𝑇 𝑞 ↔ ( ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ) ) ) |
| 44 |
43
|
simprbi |
⊢ ( 𝑝 𝑇 𝑞 → ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ) ) |
| 45 |
1 2
|
weiunlem1 |
⊢ ( 𝑞 𝑇 𝑟 ↔ ( ( 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) ) |
| 46 |
45
|
simprbi |
⊢ ( 𝑞 𝑇 𝑟 → ( ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) |
| 47 |
|
simpr2 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) |
| 48 |
8 47
|
ffvelcdmd |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝐹 ‘ 𝑞 ) ∈ 𝐴 ) |
| 49 |
8 41
|
ffvelcdmd |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝐹 ‘ 𝑟 ) ∈ 𝐴 ) |
| 50 |
|
sotr |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( ( 𝐹 ‘ 𝑝 ) ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑞 ) ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑟 ) ∈ 𝐴 ) ) → ( ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∧ ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) → ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) ) |
| 51 |
5 10 48 49 50
|
syl13anc |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∧ ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) → ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) ) |
| 52 |
|
orc |
⊢ ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) → ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) |
| 53 |
51 52
|
syl6 |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∧ ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) → ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) ) |
| 54 |
|
simprll |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) ) → ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ) |
| 55 |
|
simprr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) ) → ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) |
| 56 |
54 55
|
eqbrtrd |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) ) → ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) |
| 57 |
56
|
orcd |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) ) → ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) |
| 58 |
57
|
ex |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) → ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) ) |
| 59 |
|
simprl |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ) |
| 60 |
|
simprrl |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) |
| 61 |
59 60
|
breqtrd |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ) |
| 62 |
61
|
orcd |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) |
| 63 |
62
|
ex |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) → ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) ) |
| 64 |
|
simprll |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ) |
| 65 |
|
simprrl |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ) |
| 66 |
64 65
|
eqtrd |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ) |
| 67 |
|
simprlr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) |
| 68 |
|
simprrr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) |
| 69 |
64
|
csbeq1d |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 = ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 ) |
| 70 |
69
|
breqd |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ( 𝑞 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ↔ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) |
| 71 |
68 70
|
mpbird |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → 𝑞 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) |
| 72 |
26
|
adantr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 Po ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
| 73 |
32
|
adantr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → 𝑝 ∈ ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
| 74 |
|
id |
⊢ ( 𝑡 = 𝑞 → 𝑡 = 𝑞 ) |
| 75 |
|
fveq2 |
⊢ ( 𝑡 = 𝑞 → ( 𝐹 ‘ 𝑡 ) = ( 𝐹 ‘ 𝑞 ) ) |
| 76 |
75
|
csbeq1d |
⊢ ( 𝑡 = 𝑞 → ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 = ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ) |
| 77 |
74 76
|
eleq12d |
⊢ ( 𝑡 = 𝑞 → ( 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ↔ 𝑞 ∈ ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ) ) |
| 78 |
77 31 47
|
rspcdva |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑞 ∈ ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ) |
| 79 |
78
|
adantr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → 𝑞 ∈ ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ) |
| 80 |
64
|
csbeq1d |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 = ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝐵 ) |
| 81 |
79 80
|
eleqtrrd |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → 𝑞 ∈ ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
| 82 |
|
id |
⊢ ( 𝑡 = 𝑟 → 𝑡 = 𝑟 ) |
| 83 |
|
fveq2 |
⊢ ( 𝑡 = 𝑟 → ( 𝐹 ‘ 𝑡 ) = ( 𝐹 ‘ 𝑟 ) ) |
| 84 |
83
|
csbeq1d |
⊢ ( 𝑡 = 𝑟 → ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 = ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝐵 ) |
| 85 |
82 84
|
eleq12d |
⊢ ( 𝑡 = 𝑟 → ( 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ↔ 𝑟 ∈ ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝐵 ) ) |
| 86 |
85 31 41
|
rspcdva |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑟 ∈ ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝐵 ) |
| 87 |
86
|
adantr |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → 𝑟 ∈ ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝐵 ) |
| 88 |
66
|
csbeq1d |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 = ⦋ ( 𝐹 ‘ 𝑟 ) / 𝑥 ⦌ 𝐵 ) |
| 89 |
87 88
|
eleqtrrd |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → 𝑟 ∈ ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ) |
| 90 |
|
potr |
⊢ ( ( ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 Po ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ∧ ( 𝑝 ∈ ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ∧ 𝑞 ∈ ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ∧ 𝑟 ∈ ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝐵 ) ) → ( ( 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) → 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) |
| 91 |
72 73 81 89 90
|
syl13anc |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ( ( 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) → 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) |
| 92 |
67 71 91
|
mp2and |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) |
| 93 |
66 92
|
jca |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) |
| 94 |
93
|
olcd |
⊢ ( ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) ∧ ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) |
| 95 |
94
|
ex |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( ( ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ∧ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) → ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) ) |
| 96 |
53 58 63 95
|
ccased |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( ( ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑞 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑞 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑞 ) ) ∧ ( ( 𝐹 ‘ 𝑞 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑞 ⦋ ( 𝐹 ‘ 𝑞 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) ) |
| 97 |
44 46 96
|
syl2ani |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( ( 𝑝 𝑇 𝑞 ∧ 𝑞 𝑇 𝑟 ) → ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) ) |
| 98 |
1 2
|
weiunlem1 |
⊢ ( 𝑝 𝑇 𝑟 ↔ ( ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) ) |
| 99 |
98
|
biimpri |
⊢ ( ( ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑝 ) 𝑅 ( 𝐹 ‘ 𝑟 ) ∨ ( ( 𝐹 ‘ 𝑝 ) = ( 𝐹 ‘ 𝑟 ) ∧ 𝑝 ⦋ ( 𝐹 ‘ 𝑝 ) / 𝑥 ⦌ 𝑆 𝑟 ) ) ) → 𝑝 𝑇 𝑟 ) |
| 100 |
42 97 99
|
syl6an |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( ( 𝑝 𝑇 𝑞 ∧ 𝑞 𝑇 𝑟 ) → 𝑝 𝑇 𝑟 ) ) |
| 101 |
40 100
|
jca |
⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) ∧ ( 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ) → ( ¬ 𝑝 𝑇 𝑝 ∧ ( ( 𝑝 𝑇 𝑞 ∧ 𝑞 𝑇 𝑟 ) → 𝑝 𝑇 𝑟 ) ) ) |
| 102 |
101
|
ralrimivvva |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) → ∀ 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∀ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∀ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ( ¬ 𝑝 𝑇 𝑝 ∧ ( ( 𝑝 𝑇 𝑞 ∧ 𝑞 𝑇 𝑟 ) → 𝑝 𝑇 𝑟 ) ) ) |
| 103 |
|
df-po |
⊢ ( 𝑇 Po ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑝 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∀ 𝑞 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∀ 𝑟 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ( ¬ 𝑝 𝑇 𝑝 ∧ ( ( 𝑝 𝑇 𝑞 ∧ 𝑞 𝑇 𝑟 ) → 𝑝 𝑇 𝑟 ) ) ) |
| 104 |
102 103
|
sylibr |
⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝑆 Po 𝐵 ) → 𝑇 Po ∪ 𝑥 ∈ 𝐴 𝐵 ) |