Step |
Hyp |
Ref |
Expression |
1 |
|
wepwso.t |
⊢ 𝑇 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑧 ∈ 𝐴 ( ( 𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦 ) ) ) } |
2 |
|
wepwso.u |
⊢ 𝑈 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑧 ∈ 𝐴 ( ( 𝑥 ‘ 𝑧 ) E ( 𝑦 ‘ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑥 ‘ 𝑤 ) = ( 𝑦 ‘ 𝑤 ) ) ) } |
3 |
|
wepwso.f |
⊢ 𝐹 = ( 𝑎 ∈ ( 2o ↑m 𝐴 ) ↦ ( ◡ 𝑎 “ { 1o } ) ) |
4 |
3
|
pw2f1o2 |
⊢ ( 𝐴 ∈ V → 𝐹 : ( 2o ↑m 𝐴 ) –1-1-onto→ 𝒫 𝐴 ) |
5 |
|
fvex |
⊢ ( 𝑐 ‘ 𝑧 ) ∈ V |
6 |
5
|
epeli |
⊢ ( ( 𝑏 ‘ 𝑧 ) E ( 𝑐 ‘ 𝑧 ) ↔ ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ) |
7 |
|
elmapi |
⊢ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) → 𝑏 : 𝐴 ⟶ 2o ) |
8 |
7
|
ad2antrl |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) → 𝑏 : 𝐴 ⟶ 2o ) |
9 |
8
|
ffvelrnda |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑏 ‘ 𝑧 ) ∈ 2o ) |
10 |
|
elmapi |
⊢ ( 𝑐 ∈ ( 2o ↑m 𝐴 ) → 𝑐 : 𝐴 ⟶ 2o ) |
11 |
10
|
ad2antll |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) → 𝑐 : 𝐴 ⟶ 2o ) |
12 |
11
|
ffvelrnda |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑐 ‘ 𝑧 ) ∈ 2o ) |
13 |
|
n0i |
⊢ ( ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) → ¬ ( 𝑐 ‘ 𝑧 ) = ∅ ) |
14 |
13
|
adantl |
⊢ ( ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ) → ¬ ( 𝑐 ‘ 𝑧 ) = ∅ ) |
15 |
|
elpri |
⊢ ( ( 𝑐 ‘ 𝑧 ) ∈ { ∅ , 1o } → ( ( 𝑐 ‘ 𝑧 ) = ∅ ∨ ( 𝑐 ‘ 𝑧 ) = 1o ) ) |
16 |
|
df2o3 |
⊢ 2o = { ∅ , 1o } |
17 |
15 16
|
eleq2s |
⊢ ( ( 𝑐 ‘ 𝑧 ) ∈ 2o → ( ( 𝑐 ‘ 𝑧 ) = ∅ ∨ ( 𝑐 ‘ 𝑧 ) = 1o ) ) |
18 |
17
|
ad2antlr |
⊢ ( ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ) → ( ( 𝑐 ‘ 𝑧 ) = ∅ ∨ ( 𝑐 ‘ 𝑧 ) = 1o ) ) |
19 |
|
orel1 |
⊢ ( ¬ ( 𝑐 ‘ 𝑧 ) = ∅ → ( ( ( 𝑐 ‘ 𝑧 ) = ∅ ∨ ( 𝑐 ‘ 𝑧 ) = 1o ) → ( 𝑐 ‘ 𝑧 ) = 1o ) ) |
20 |
14 18 19
|
sylc |
⊢ ( ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ) → ( 𝑐 ‘ 𝑧 ) = 1o ) |
21 |
|
1on |
⊢ 1o ∈ On |
22 |
21
|
onirri |
⊢ ¬ 1o ∈ 1o |
23 |
|
eleq12 |
⊢ ( ( ( 𝑏 ‘ 𝑧 ) = 1o ∧ ( 𝑐 ‘ 𝑧 ) = 1o ) → ( ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ↔ 1o ∈ 1o ) ) |
24 |
23
|
biimpd |
⊢ ( ( ( 𝑏 ‘ 𝑧 ) = 1o ∧ ( 𝑐 ‘ 𝑧 ) = 1o ) → ( ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) → 1o ∈ 1o ) ) |
25 |
24
|
expcom |
⊢ ( ( 𝑐 ‘ 𝑧 ) = 1o → ( ( 𝑏 ‘ 𝑧 ) = 1o → ( ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) → 1o ∈ 1o ) ) ) |
26 |
25
|
com3r |
⊢ ( ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) → ( ( 𝑐 ‘ 𝑧 ) = 1o → ( ( 𝑏 ‘ 𝑧 ) = 1o → 1o ∈ 1o ) ) ) |
27 |
26
|
imp |
⊢ ( ( ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ∧ ( 𝑐 ‘ 𝑧 ) = 1o ) → ( ( 𝑏 ‘ 𝑧 ) = 1o → 1o ∈ 1o ) ) |
28 |
27
|
adantll |
⊢ ( ( ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ) ∧ ( 𝑐 ‘ 𝑧 ) = 1o ) → ( ( 𝑏 ‘ 𝑧 ) = 1o → 1o ∈ 1o ) ) |
29 |
22 28
|
mtoi |
⊢ ( ( ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ) ∧ ( 𝑐 ‘ 𝑧 ) = 1o ) → ¬ ( 𝑏 ‘ 𝑧 ) = 1o ) |
30 |
20 29
|
mpdan |
⊢ ( ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ) → ¬ ( 𝑏 ‘ 𝑧 ) = 1o ) |
31 |
20 30
|
jca |
⊢ ( ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ) → ( ( 𝑐 ‘ 𝑧 ) = 1o ∧ ¬ ( 𝑏 ‘ 𝑧 ) = 1o ) ) |
32 |
|
elpri |
⊢ ( ( 𝑏 ‘ 𝑧 ) ∈ { ∅ , 1o } → ( ( 𝑏 ‘ 𝑧 ) = ∅ ∨ ( 𝑏 ‘ 𝑧 ) = 1o ) ) |
33 |
32 16
|
eleq2s |
⊢ ( ( 𝑏 ‘ 𝑧 ) ∈ 2o → ( ( 𝑏 ‘ 𝑧 ) = ∅ ∨ ( 𝑏 ‘ 𝑧 ) = 1o ) ) |
34 |
33
|
adantr |
⊢ ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) → ( ( 𝑏 ‘ 𝑧 ) = ∅ ∨ ( 𝑏 ‘ 𝑧 ) = 1o ) ) |
35 |
|
orel2 |
⊢ ( ¬ ( 𝑏 ‘ 𝑧 ) = 1o → ( ( ( 𝑏 ‘ 𝑧 ) = ∅ ∨ ( 𝑏 ‘ 𝑧 ) = 1o ) → ( 𝑏 ‘ 𝑧 ) = ∅ ) ) |
36 |
34 35
|
mpan9 |
⊢ ( ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) ∧ ¬ ( 𝑏 ‘ 𝑧 ) = 1o ) → ( 𝑏 ‘ 𝑧 ) = ∅ ) |
37 |
36
|
adantrl |
⊢ ( ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) ∧ ( ( 𝑐 ‘ 𝑧 ) = 1o ∧ ¬ ( 𝑏 ‘ 𝑧 ) = 1o ) ) → ( 𝑏 ‘ 𝑧 ) = ∅ ) |
38 |
|
0lt1o |
⊢ ∅ ∈ 1o |
39 |
37 38
|
eqeltrdi |
⊢ ( ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) ∧ ( ( 𝑐 ‘ 𝑧 ) = 1o ∧ ¬ ( 𝑏 ‘ 𝑧 ) = 1o ) ) → ( 𝑏 ‘ 𝑧 ) ∈ 1o ) |
40 |
|
simprl |
⊢ ( ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) ∧ ( ( 𝑐 ‘ 𝑧 ) = 1o ∧ ¬ ( 𝑏 ‘ 𝑧 ) = 1o ) ) → ( 𝑐 ‘ 𝑧 ) = 1o ) |
41 |
39 40
|
eleqtrrd |
⊢ ( ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) ∧ ( ( 𝑐 ‘ 𝑧 ) = 1o ∧ ¬ ( 𝑏 ‘ 𝑧 ) = 1o ) ) → ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ) |
42 |
31 41
|
impbida |
⊢ ( ( ( 𝑏 ‘ 𝑧 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑧 ) ∈ 2o ) → ( ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ↔ ( ( 𝑐 ‘ 𝑧 ) = 1o ∧ ¬ ( 𝑏 ‘ 𝑧 ) = 1o ) ) ) |
43 |
9 12 42
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ↔ ( ( 𝑐 ‘ 𝑧 ) = 1o ∧ ¬ ( 𝑏 ‘ 𝑧 ) = 1o ) ) ) |
44 |
|
simplrr |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑧 ∈ 𝐴 ) → 𝑐 ∈ ( 2o ↑m 𝐴 ) ) |
45 |
3
|
pw2f1o2val2 |
⊢ ( ( 𝑐 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑧 ∈ ( 𝐹 ‘ 𝑐 ) ↔ ( 𝑐 ‘ 𝑧 ) = 1o ) ) |
46 |
44 45
|
sylancom |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑧 ∈ ( 𝐹 ‘ 𝑐 ) ↔ ( 𝑐 ‘ 𝑧 ) = 1o ) ) |
47 |
|
simplrl |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑧 ∈ 𝐴 ) → 𝑏 ∈ ( 2o ↑m 𝐴 ) ) |
48 |
3
|
pw2f1o2val2 |
⊢ ( ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ↔ ( 𝑏 ‘ 𝑧 ) = 1o ) ) |
49 |
47 48
|
sylancom |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑧 ∈ 𝐴 ) → ( 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ↔ ( 𝑏 ‘ 𝑧 ) = 1o ) ) |
50 |
49
|
notbid |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑧 ∈ 𝐴 ) → ( ¬ 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ↔ ¬ ( 𝑏 ‘ 𝑧 ) = 1o ) ) |
51 |
46 50
|
anbi12d |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑧 ∈ ( 𝐹 ‘ 𝑐 ) ∧ ¬ 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ) ↔ ( ( 𝑐 ‘ 𝑧 ) = 1o ∧ ¬ ( 𝑏 ‘ 𝑧 ) = 1o ) ) ) |
52 |
43 51
|
bitr4d |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑏 ‘ 𝑧 ) ∈ ( 𝑐 ‘ 𝑧 ) ↔ ( 𝑧 ∈ ( 𝐹 ‘ 𝑐 ) ∧ ¬ 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ) ) ) |
53 |
6 52
|
syl5bb |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑧 ∈ 𝐴 ) → ( ( 𝑏 ‘ 𝑧 ) E ( 𝑐 ‘ 𝑧 ) ↔ ( 𝑧 ∈ ( 𝐹 ‘ 𝑐 ) ∧ ¬ 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ) ) ) |
54 |
8
|
ffvelrnda |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑤 ∈ 𝐴 ) → ( 𝑏 ‘ 𝑤 ) ∈ 2o ) |
55 |
11
|
ffvelrnda |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑤 ∈ 𝐴 ) → ( 𝑐 ‘ 𝑤 ) ∈ 2o ) |
56 |
|
eqeq1 |
⊢ ( ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) → ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) |
57 |
|
simplr |
⊢ ( ( ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑤 ) = ∅ ) ∧ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) → ( 𝑏 ‘ 𝑤 ) = ∅ ) |
58 |
|
1n0 |
⊢ 1o ≠ ∅ |
59 |
58
|
nesymi |
⊢ ¬ ∅ = 1o |
60 |
|
eqeq1 |
⊢ ( ( 𝑏 ‘ 𝑤 ) = ∅ → ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ∅ = 1o ) ) |
61 |
59 60
|
mtbiri |
⊢ ( ( 𝑏 ‘ 𝑤 ) = ∅ → ¬ ( 𝑏 ‘ 𝑤 ) = 1o ) |
62 |
61
|
ad2antlr |
⊢ ( ( ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑤 ) = ∅ ) ∧ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) → ¬ ( 𝑏 ‘ 𝑤 ) = 1o ) |
63 |
|
simpr |
⊢ ( ( ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑤 ) = ∅ ) ∧ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) → ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) |
64 |
62 63
|
mtbid |
⊢ ( ( ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑤 ) = ∅ ) ∧ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) → ¬ ( 𝑐 ‘ 𝑤 ) = 1o ) |
65 |
|
elpri |
⊢ ( ( 𝑐 ‘ 𝑤 ) ∈ { ∅ , 1o } → ( ( 𝑐 ‘ 𝑤 ) = ∅ ∨ ( 𝑐 ‘ 𝑤 ) = 1o ) ) |
66 |
65 16
|
eleq2s |
⊢ ( ( 𝑐 ‘ 𝑤 ) ∈ 2o → ( ( 𝑐 ‘ 𝑤 ) = ∅ ∨ ( 𝑐 ‘ 𝑤 ) = 1o ) ) |
67 |
66
|
ad3antlr |
⊢ ( ( ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑤 ) = ∅ ) ∧ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) → ( ( 𝑐 ‘ 𝑤 ) = ∅ ∨ ( 𝑐 ‘ 𝑤 ) = 1o ) ) |
68 |
|
orel2 |
⊢ ( ¬ ( 𝑐 ‘ 𝑤 ) = 1o → ( ( ( 𝑐 ‘ 𝑤 ) = ∅ ∨ ( 𝑐 ‘ 𝑤 ) = 1o ) → ( 𝑐 ‘ 𝑤 ) = ∅ ) ) |
69 |
64 67 68
|
sylc |
⊢ ( ( ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑤 ) = ∅ ) ∧ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) → ( 𝑐 ‘ 𝑤 ) = ∅ ) |
70 |
57 69
|
eqtr4d |
⊢ ( ( ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑤 ) = ∅ ) ∧ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) |
71 |
70
|
ex |
⊢ ( ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑤 ) = ∅ ) → ( ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ) |
72 |
|
simplr |
⊢ ( ( ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑤 ) = 1o ) ∧ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) → ( 𝑏 ‘ 𝑤 ) = 1o ) |
73 |
|
simpr |
⊢ ( ( ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑤 ) = 1o ) ∧ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) → ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) |
74 |
72 73
|
mpbid |
⊢ ( ( ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑤 ) = 1o ) ∧ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) → ( 𝑐 ‘ 𝑤 ) = 1o ) |
75 |
72 74
|
eqtr4d |
⊢ ( ( ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑤 ) = 1o ) ∧ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) |
76 |
75
|
ex |
⊢ ( ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) ∧ ( 𝑏 ‘ 𝑤 ) = 1o ) → ( ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ) |
77 |
|
elpri |
⊢ ( ( 𝑏 ‘ 𝑤 ) ∈ { ∅ , 1o } → ( ( 𝑏 ‘ 𝑤 ) = ∅ ∨ ( 𝑏 ‘ 𝑤 ) = 1o ) ) |
78 |
77 16
|
eleq2s |
⊢ ( ( 𝑏 ‘ 𝑤 ) ∈ 2o → ( ( 𝑏 ‘ 𝑤 ) = ∅ ∨ ( 𝑏 ‘ 𝑤 ) = 1o ) ) |
79 |
78
|
adantr |
⊢ ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) → ( ( 𝑏 ‘ 𝑤 ) = ∅ ∨ ( 𝑏 ‘ 𝑤 ) = 1o ) ) |
80 |
71 76 79
|
mpjaodan |
⊢ ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) → ( ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ) |
81 |
56 80
|
impbid2 |
⊢ ( ( ( 𝑏 ‘ 𝑤 ) ∈ 2o ∧ ( 𝑐 ‘ 𝑤 ) ∈ 2o ) → ( ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ↔ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) ) |
82 |
54 55 81
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ↔ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) ) |
83 |
|
simplrl |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑤 ∈ 𝐴 ) → 𝑏 ∈ ( 2o ↑m 𝐴 ) ) |
84 |
3
|
pw2f1o2val2 |
⊢ ( ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑤 ∈ 𝐴 ) → ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ ( 𝑏 ‘ 𝑤 ) = 1o ) ) |
85 |
83 84
|
sylancom |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑤 ∈ 𝐴 ) → ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ ( 𝑏 ‘ 𝑤 ) = 1o ) ) |
86 |
|
simplrr |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑤 ∈ 𝐴 ) → 𝑐 ∈ ( 2o ↑m 𝐴 ) ) |
87 |
3
|
pw2f1o2val2 |
⊢ ( ( 𝑐 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑤 ∈ 𝐴 ) → ( 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) |
88 |
86 87
|
sylancom |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑤 ∈ 𝐴 ) → ( 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) |
89 |
85 88
|
bibi12d |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ↔ ( ( 𝑏 ‘ 𝑤 ) = 1o ↔ ( 𝑐 ‘ 𝑤 ) = 1o ) ) ) |
90 |
82 89
|
bitr4d |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ↔ ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ) ) |
91 |
90
|
imbi2d |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝑤 𝑅 𝑧 → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ↔ ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ) ) ) |
92 |
91
|
ralbidva |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) → ( ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ↔ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ) ) ) |
93 |
92
|
adantr |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑧 ∈ 𝐴 ) → ( ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ↔ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ) ) ) |
94 |
53 93
|
anbi12d |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) ∧ 𝑧 ∈ 𝐴 ) → ( ( ( 𝑏 ‘ 𝑧 ) E ( 𝑐 ‘ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ) ↔ ( ( 𝑧 ∈ ( 𝐹 ‘ 𝑐 ) ∧ ¬ 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ) ) ) ) |
95 |
94
|
rexbidva |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) → ( ∃ 𝑧 ∈ 𝐴 ( ( 𝑏 ‘ 𝑧 ) E ( 𝑐 ‘ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ) ↔ ∃ 𝑧 ∈ 𝐴 ( ( 𝑧 ∈ ( 𝐹 ‘ 𝑐 ) ∧ ¬ 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ) ) ) ) |
96 |
|
vex |
⊢ 𝑏 ∈ V |
97 |
|
vex |
⊢ 𝑐 ∈ V |
98 |
|
fveq1 |
⊢ ( 𝑥 = 𝑏 → ( 𝑥 ‘ 𝑧 ) = ( 𝑏 ‘ 𝑧 ) ) |
99 |
|
fveq1 |
⊢ ( 𝑦 = 𝑐 → ( 𝑦 ‘ 𝑧 ) = ( 𝑐 ‘ 𝑧 ) ) |
100 |
98 99
|
breqan12d |
⊢ ( ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑐 ) → ( ( 𝑥 ‘ 𝑧 ) E ( 𝑦 ‘ 𝑧 ) ↔ ( 𝑏 ‘ 𝑧 ) E ( 𝑐 ‘ 𝑧 ) ) ) |
101 |
|
fveq1 |
⊢ ( 𝑥 = 𝑏 → ( 𝑥 ‘ 𝑤 ) = ( 𝑏 ‘ 𝑤 ) ) |
102 |
|
fveq1 |
⊢ ( 𝑦 = 𝑐 → ( 𝑦 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) |
103 |
101 102
|
eqeqan12d |
⊢ ( ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑐 ) → ( ( 𝑥 ‘ 𝑤 ) = ( 𝑦 ‘ 𝑤 ) ↔ ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ) |
104 |
103
|
imbi2d |
⊢ ( ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑐 ) → ( ( 𝑤 𝑅 𝑧 → ( 𝑥 ‘ 𝑤 ) = ( 𝑦 ‘ 𝑤 ) ) ↔ ( 𝑤 𝑅 𝑧 → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ) ) |
105 |
104
|
ralbidv |
⊢ ( ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑐 ) → ( ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑥 ‘ 𝑤 ) = ( 𝑦 ‘ 𝑤 ) ) ↔ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ) ) |
106 |
100 105
|
anbi12d |
⊢ ( ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑐 ) → ( ( ( 𝑥 ‘ 𝑧 ) E ( 𝑦 ‘ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑥 ‘ 𝑤 ) = ( 𝑦 ‘ 𝑤 ) ) ) ↔ ( ( 𝑏 ‘ 𝑧 ) E ( 𝑐 ‘ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ) ) ) |
107 |
106
|
rexbidv |
⊢ ( ( 𝑥 = 𝑏 ∧ 𝑦 = 𝑐 ) → ( ∃ 𝑧 ∈ 𝐴 ( ( 𝑥 ‘ 𝑧 ) E ( 𝑦 ‘ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑥 ‘ 𝑤 ) = ( 𝑦 ‘ 𝑤 ) ) ) ↔ ∃ 𝑧 ∈ 𝐴 ( ( 𝑏 ‘ 𝑧 ) E ( 𝑐 ‘ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ) ) ) |
108 |
96 97 107 2
|
braba |
⊢ ( 𝑏 𝑈 𝑐 ↔ ∃ 𝑧 ∈ 𝐴 ( ( 𝑏 ‘ 𝑧 ) E ( 𝑐 ‘ 𝑧 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑏 ‘ 𝑤 ) = ( 𝑐 ‘ 𝑤 ) ) ) ) |
109 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑏 ) ∈ V |
110 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑐 ) ∈ V |
111 |
|
eleq2 |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑐 ) → ( 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ( 𝐹 ‘ 𝑐 ) ) ) |
112 |
|
eleq2 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑏 ) → ( 𝑧 ∈ 𝑥 ↔ 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ) ) |
113 |
112
|
notbid |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑏 ) → ( ¬ 𝑧 ∈ 𝑥 ↔ ¬ 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ) ) |
114 |
111 113
|
bi2anan9r |
⊢ ( ( 𝑥 = ( 𝐹 ‘ 𝑏 ) ∧ 𝑦 = ( 𝐹 ‘ 𝑐 ) ) → ( ( 𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥 ) ↔ ( 𝑧 ∈ ( 𝐹 ‘ 𝑐 ) ∧ ¬ 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ) ) ) |
115 |
|
eleq2 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑏 ) → ( 𝑤 ∈ 𝑥 ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ) ) |
116 |
|
eleq2 |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑐 ) → ( 𝑤 ∈ 𝑦 ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ) |
117 |
115 116
|
bi2bian9 |
⊢ ( ( 𝑥 = ( 𝐹 ‘ 𝑏 ) ∧ 𝑦 = ( 𝐹 ‘ 𝑐 ) ) → ( ( 𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦 ) ↔ ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ) ) |
118 |
117
|
imbi2d |
⊢ ( ( 𝑥 = ( 𝐹 ‘ 𝑏 ) ∧ 𝑦 = ( 𝐹 ‘ 𝑐 ) ) → ( ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦 ) ) ↔ ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ) ) ) |
119 |
118
|
ralbidv |
⊢ ( ( 𝑥 = ( 𝐹 ‘ 𝑏 ) ∧ 𝑦 = ( 𝐹 ‘ 𝑐 ) ) → ( ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦 ) ) ↔ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ) ) ) |
120 |
114 119
|
anbi12d |
⊢ ( ( 𝑥 = ( 𝐹 ‘ 𝑏 ) ∧ 𝑦 = ( 𝐹 ‘ 𝑐 ) ) → ( ( ( 𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦 ) ) ) ↔ ( ( 𝑧 ∈ ( 𝐹 ‘ 𝑐 ) ∧ ¬ 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ) ) ) ) |
121 |
120
|
rexbidv |
⊢ ( ( 𝑥 = ( 𝐹 ‘ 𝑏 ) ∧ 𝑦 = ( 𝐹 ‘ 𝑐 ) ) → ( ∃ 𝑧 ∈ 𝐴 ( ( 𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥 ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦 ) ) ) ↔ ∃ 𝑧 ∈ 𝐴 ( ( 𝑧 ∈ ( 𝐹 ‘ 𝑐 ) ∧ ¬ 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ) ) ) ) |
122 |
109 110 121 1
|
braba |
⊢ ( ( 𝐹 ‘ 𝑏 ) 𝑇 ( 𝐹 ‘ 𝑐 ) ↔ ∃ 𝑧 ∈ 𝐴 ( ( 𝑧 ∈ ( 𝐹 ‘ 𝑐 ) ∧ ¬ 𝑧 ∈ ( 𝐹 ‘ 𝑏 ) ) ∧ ∀ 𝑤 ∈ 𝐴 ( 𝑤 𝑅 𝑧 → ( 𝑤 ∈ ( 𝐹 ‘ 𝑏 ) ↔ 𝑤 ∈ ( 𝐹 ‘ 𝑐 ) ) ) ) ) |
123 |
95 108 122
|
3bitr4g |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝑏 ∈ ( 2o ↑m 𝐴 ) ∧ 𝑐 ∈ ( 2o ↑m 𝐴 ) ) ) → ( 𝑏 𝑈 𝑐 ↔ ( 𝐹 ‘ 𝑏 ) 𝑇 ( 𝐹 ‘ 𝑐 ) ) ) |
124 |
123
|
ralrimivva |
⊢ ( 𝐴 ∈ V → ∀ 𝑏 ∈ ( 2o ↑m 𝐴 ) ∀ 𝑐 ∈ ( 2o ↑m 𝐴 ) ( 𝑏 𝑈 𝑐 ↔ ( 𝐹 ‘ 𝑏 ) 𝑇 ( 𝐹 ‘ 𝑐 ) ) ) |
125 |
|
df-isom |
⊢ ( 𝐹 Isom 𝑈 , 𝑇 ( ( 2o ↑m 𝐴 ) , 𝒫 𝐴 ) ↔ ( 𝐹 : ( 2o ↑m 𝐴 ) –1-1-onto→ 𝒫 𝐴 ∧ ∀ 𝑏 ∈ ( 2o ↑m 𝐴 ) ∀ 𝑐 ∈ ( 2o ↑m 𝐴 ) ( 𝑏 𝑈 𝑐 ↔ ( 𝐹 ‘ 𝑏 ) 𝑇 ( 𝐹 ‘ 𝑐 ) ) ) ) |
126 |
4 124 125
|
sylanbrc |
⊢ ( 𝐴 ∈ V → 𝐹 Isom 𝑈 , 𝑇 ( ( 2o ↑m 𝐴 ) , 𝒫 𝐴 ) ) |