Step |
Hyp |
Ref |
Expression |
1 |
|
isocnv3.1 |
⊢ 𝐶 = ( ( 𝐴 × 𝐴 ) ∖ 𝑅 ) |
2 |
|
isocnv3.2 |
⊢ 𝐷 = ( ( 𝐵 × 𝐵 ) ∖ 𝑆 ) |
3 |
|
brxp |
⊢ ( 𝑥 ( 𝐴 × 𝐴 ) 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) |
4 |
1
|
breqi |
⊢ ( 𝑥 𝐶 𝑦 ↔ 𝑥 ( ( 𝐴 × 𝐴 ) ∖ 𝑅 ) 𝑦 ) |
5 |
|
brdif |
⊢ ( 𝑥 ( ( 𝐴 × 𝐴 ) ∖ 𝑅 ) 𝑦 ↔ ( 𝑥 ( 𝐴 × 𝐴 ) 𝑦 ∧ ¬ 𝑥 𝑅 𝑦 ) ) |
6 |
4 5
|
bitri |
⊢ ( 𝑥 𝐶 𝑦 ↔ ( 𝑥 ( 𝐴 × 𝐴 ) 𝑦 ∧ ¬ 𝑥 𝑅 𝑦 ) ) |
7 |
6
|
baib |
⊢ ( 𝑥 ( 𝐴 × 𝐴 ) 𝑦 → ( 𝑥 𝐶 𝑦 ↔ ¬ 𝑥 𝑅 𝑦 ) ) |
8 |
3 7
|
sylbir |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 𝐶 𝑦 ↔ ¬ 𝑥 𝑅 𝑦 ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 𝐶 𝑦 ↔ ¬ 𝑥 𝑅 𝑦 ) ) |
10 |
|
f1of |
⊢ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 → 𝐻 : 𝐴 ⟶ 𝐵 ) |
11 |
|
ffvelrn |
⊢ ( ( 𝐻 : 𝐴 ⟶ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐻 ‘ 𝑥 ) ∈ 𝐵 ) |
12 |
|
ffvelrn |
⊢ ( ( 𝐻 : 𝐴 ⟶ 𝐵 ∧ 𝑦 ∈ 𝐴 ) → ( 𝐻 ‘ 𝑦 ) ∈ 𝐵 ) |
13 |
11 12
|
anim12dan |
⊢ ( ( 𝐻 : 𝐴 ⟶ 𝐵 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝐻 ‘ 𝑥 ) ∈ 𝐵 ∧ ( 𝐻 ‘ 𝑦 ) ∈ 𝐵 ) ) |
14 |
|
brxp |
⊢ ( ( 𝐻 ‘ 𝑥 ) ( 𝐵 × 𝐵 ) ( 𝐻 ‘ 𝑦 ) ↔ ( ( 𝐻 ‘ 𝑥 ) ∈ 𝐵 ∧ ( 𝐻 ‘ 𝑦 ) ∈ 𝐵 ) ) |
15 |
13 14
|
sylibr |
⊢ ( ( 𝐻 : 𝐴 ⟶ 𝐵 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝐻 ‘ 𝑥 ) ( 𝐵 × 𝐵 ) ( 𝐻 ‘ 𝑦 ) ) |
16 |
10 15
|
sylan |
⊢ ( ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝐻 ‘ 𝑥 ) ( 𝐵 × 𝐵 ) ( 𝐻 ‘ 𝑦 ) ) |
17 |
2
|
breqi |
⊢ ( ( 𝐻 ‘ 𝑥 ) 𝐷 ( 𝐻 ‘ 𝑦 ) ↔ ( 𝐻 ‘ 𝑥 ) ( ( 𝐵 × 𝐵 ) ∖ 𝑆 ) ( 𝐻 ‘ 𝑦 ) ) |
18 |
|
brdif |
⊢ ( ( 𝐻 ‘ 𝑥 ) ( ( 𝐵 × 𝐵 ) ∖ 𝑆 ) ( 𝐻 ‘ 𝑦 ) ↔ ( ( 𝐻 ‘ 𝑥 ) ( 𝐵 × 𝐵 ) ( 𝐻 ‘ 𝑦 ) ∧ ¬ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) |
19 |
17 18
|
bitri |
⊢ ( ( 𝐻 ‘ 𝑥 ) 𝐷 ( 𝐻 ‘ 𝑦 ) ↔ ( ( 𝐻 ‘ 𝑥 ) ( 𝐵 × 𝐵 ) ( 𝐻 ‘ 𝑦 ) ∧ ¬ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) |
20 |
19
|
baib |
⊢ ( ( 𝐻 ‘ 𝑥 ) ( 𝐵 × 𝐵 ) ( 𝐻 ‘ 𝑦 ) → ( ( 𝐻 ‘ 𝑥 ) 𝐷 ( 𝐻 ‘ 𝑦 ) ↔ ¬ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) |
21 |
16 20
|
syl |
⊢ ( ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝐻 ‘ 𝑥 ) 𝐷 ( 𝐻 ‘ 𝑦 ) ↔ ¬ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) |
22 |
9 21
|
bibi12d |
⊢ ( ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝑥 𝐶 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝐷 ( 𝐻 ‘ 𝑦 ) ) ↔ ( ¬ 𝑥 𝑅 𝑦 ↔ ¬ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) ) |
23 |
|
notbi |
⊢ ( ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ↔ ( ¬ 𝑥 𝑅 𝑦 ↔ ¬ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) |
24 |
22 23
|
syl6rbbr |
⊢ ( ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ↔ ( 𝑥 𝐶 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝐷 ( 𝐻 ‘ 𝑦 ) ) ) ) |
25 |
24
|
2ralbidva |
⊢ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝐶 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝐷 ( 𝐻 ‘ 𝑦 ) ) ) ) |
26 |
25
|
pm5.32i |
⊢ ( ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) ↔ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝐶 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝐷 ( 𝐻 ‘ 𝑦 ) ) ) ) |
27 |
|
df-isom |
⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝑆 ( 𝐻 ‘ 𝑦 ) ) ) ) |
28 |
|
df-isom |
⊢ ( 𝐻 Isom 𝐶 , 𝐷 ( 𝐴 , 𝐵 ) ↔ ( 𝐻 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝐶 𝑦 ↔ ( 𝐻 ‘ 𝑥 ) 𝐷 ( 𝐻 ‘ 𝑦 ) ) ) ) |
29 |
26 27 28
|
3bitr4i |
⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ 𝐻 Isom 𝐶 , 𝐷 ( 𝐴 , 𝐵 ) ) |