Step |
Hyp |
Ref |
Expression |
1 |
|
f1of1 |
⊢ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 → 𝐻 : 𝐴 –1-1→ 𝐵 ) |
2 |
|
f1fveq |
⊢ ( ( 𝐻 : 𝐴 –1-1→ 𝐵 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝐻 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑦 ) ↔ 𝑥 = 𝑦 ) ) |
3 |
1 2
|
sylan |
⊢ ( ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝐻 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑦 ) ↔ 𝑥 = 𝑦 ) ) |
4 |
|
fvex |
⊢ ( 𝐻 ‘ 𝑦 ) ∈ V |
5 |
4
|
ideq |
⊢ ( ( 𝐻 ‘ 𝑥 ) I ( 𝐻 ‘ 𝑦 ) ↔ ( 𝐻 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑦 ) ) |
6 |
5
|
a1i |
⊢ ( ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝐻 ‘ 𝑥 ) I ( 𝐻 ‘ 𝑦 ) ↔ ( 𝐻 ‘ 𝑥 ) = ( 𝐻 ‘ 𝑦 ) ) ) |
7 |
|
ideqg |
⊢ ( 𝑦 ∈ 𝐴 → ( 𝑥 I 𝑦 ↔ 𝑥 = 𝑦 ) ) |
8 |
7
|
ad2antll |
⊢ ( ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 I 𝑦 ↔ 𝑥 = 𝑦 ) ) |
9 |
3 6 8
|
3bitr4rd |
⊢ ( ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 I 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) I ( 𝐻 ‘ 𝑦 ) ) ) |
10 |
9
|
ralrimivva |
⊢ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 I 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) I ( 𝐻 ‘ 𝑦 ) ) ) |
11 |
10
|
pm4.71i |
⊢ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ↔ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 I 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) I ( 𝐻 ‘ 𝑦 ) ) ) ) |
12 |
|
df-isom |
⊢ ( 𝐻 Isom I , I ( 𝐴 , 𝐵 ) ↔ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 I 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) I ( 𝐻 ‘ 𝑦 ) ) ) ) |
13 |
11 12
|
bitr4i |
⊢ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ↔ 𝐻 Isom I , I ( 𝐴 , 𝐵 ) ) |