Step |
Hyp |
Ref |
Expression |
1 |
|
ax-hilex |
⊢ ℋ ∈ V |
2 |
|
fex2 |
⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ ℋ ∈ V ∧ ℋ ∈ V ) → 𝑇 ∈ V ) |
3 |
1 1 2
|
mp3an23 |
⊢ ( 𝑇 : ℋ ⟶ ℋ → 𝑇 ∈ V ) |
4 |
|
feq1 |
⊢ ( 𝑡 = 𝑇 → ( 𝑡 : ℋ ⟶ ℋ ↔ 𝑇 : ℋ ⟶ ℋ ) ) |
5 |
|
fveq1 |
⊢ ( 𝑡 = 𝑇 → ( 𝑡 ‘ 𝑦 ) = ( 𝑇 ‘ 𝑦 ) ) |
6 |
5
|
oveq2d |
⊢ ( 𝑡 = 𝑇 → ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) ) |
7 |
6
|
eqeq1d |
⊢ ( 𝑡 = 𝑇 → ( ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ↔ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) |
8 |
7
|
2ralbidv |
⊢ ( 𝑡 = 𝑇 → ( ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ↔ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) |
9 |
4 8
|
3anbi13d |
⊢ ( 𝑡 = 𝑇 → ( ( 𝑡 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) ) |
10 |
|
3anass |
⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ( 𝑇 : ℋ ⟶ ℋ ∧ ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) ) |
11 |
9 10
|
syl6bb |
⊢ ( 𝑡 = 𝑇 → ( ( 𝑡 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ( 𝑇 : ℋ ⟶ ℋ ∧ ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) ) ) |
12 |
11
|
exbidv |
⊢ ( 𝑡 = 𝑇 → ( ∃ 𝑢 ( 𝑡 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ∃ 𝑢 ( 𝑇 : ℋ ⟶ ℋ ∧ ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) ) ) |
13 |
|
19.42v |
⊢ ( ∃ 𝑢 ( 𝑇 : ℋ ⟶ ℋ ∧ ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) ↔ ( 𝑇 : ℋ ⟶ ℋ ∧ ∃ 𝑢 ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) ) |
14 |
12 13
|
syl6bb |
⊢ ( 𝑡 = 𝑇 → ( ∃ 𝑢 ( 𝑡 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ( 𝑇 : ℋ ⟶ ℋ ∧ ∃ 𝑢 ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) ) ) |
15 |
|
dfadj2 |
⊢ adjℎ = { 〈 𝑡 , 𝑢 〉 ∣ ( 𝑡 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) } |
16 |
15
|
dmeqi |
⊢ dom adjℎ = dom { 〈 𝑡 , 𝑢 〉 ∣ ( 𝑡 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) } |
17 |
|
dmopab |
⊢ dom { 〈 𝑡 , 𝑢 〉 ∣ ( 𝑡 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) } = { 𝑡 ∣ ∃ 𝑢 ( 𝑡 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) } |
18 |
16 17
|
eqtri |
⊢ dom adjℎ = { 𝑡 ∣ ∃ 𝑢 ( 𝑡 : ℋ ⟶ ℋ ∧ 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑡 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) } |
19 |
14 18
|
elab2g |
⊢ ( 𝑇 ∈ V → ( 𝑇 ∈ dom adjℎ ↔ ( 𝑇 : ℋ ⟶ ℋ ∧ ∃ 𝑢 ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) ) ) |
20 |
19
|
baibd |
⊢ ( ( 𝑇 ∈ V ∧ 𝑇 : ℋ ⟶ ℋ ) → ( 𝑇 ∈ dom adjℎ ↔ ∃ 𝑢 ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) ) |
21 |
3 20
|
mpancom |
⊢ ( 𝑇 : ℋ ⟶ ℋ → ( 𝑇 ∈ dom adjℎ ↔ ∃ 𝑢 ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) ) |
22 |
|
df-reu |
⊢ ( ∃! 𝑢 ∈ ( ℋ ↑m ℋ ) ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ↔ ∃! 𝑢 ( 𝑢 ∈ ( ℋ ↑m ℋ ) ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) |
23 |
1 1
|
elmap |
⊢ ( 𝑢 ∈ ( ℋ ↑m ℋ ) ↔ 𝑢 : ℋ ⟶ ℋ ) |
24 |
23
|
anbi1i |
⊢ ( ( 𝑢 ∈ ( ℋ ↑m ℋ ) ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) |
25 |
24
|
eubii |
⊢ ( ∃! 𝑢 ( 𝑢 ∈ ( ℋ ↑m ℋ ) ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ∃! 𝑢 ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) |
26 |
|
adjmo |
⊢ ∃* 𝑢 ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) |
27 |
|
df-eu |
⊢ ( ∃! 𝑢 ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ( ∃ 𝑢 ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ∧ ∃* 𝑢 ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) ) |
28 |
26 27
|
mpbiran2 |
⊢ ( ∃! 𝑢 ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ↔ ∃ 𝑢 ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) |
29 |
22 25 28
|
3bitri |
⊢ ( ∃! 𝑢 ∈ ( ℋ ↑m ℋ ) ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ↔ ∃ 𝑢 ( 𝑢 : ℋ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) |
30 |
21 29
|
bitr4di |
⊢ ( 𝑇 : ℋ ⟶ ℋ → ( 𝑇 ∈ dom adjℎ ↔ ∃! 𝑢 ∈ ( ℋ ↑m ℋ ) ∀ 𝑥 ∈ ℋ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih ( 𝑇 ‘ 𝑦 ) ) = ( ( 𝑢 ‘ 𝑥 ) ·ih 𝑦 ) ) ) |