Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ 𝐴 ) |
2 |
1
|
snssd |
⊢ ( ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴 ) → { 𝐵 } ⊆ 𝐴 ) |
3 |
|
df-ss |
⊢ ( { 𝐵 } ⊆ 𝐴 ↔ ( { 𝐵 } ∩ 𝐴 ) = { 𝐵 } ) |
4 |
2 3
|
sylib |
⊢ ( ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( { 𝐵 } ∩ 𝐴 ) = { 𝐵 } ) |
5 |
4
|
imaeq2d |
⊢ ( ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( 𝑅 “ ( { 𝐵 } ∩ 𝐴 ) ) = ( 𝑅 “ { 𝐵 } ) ) |
6 |
5
|
ineq1d |
⊢ ( ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( ( 𝑅 “ ( { 𝐵 } ∩ 𝐴 ) ) ∩ 𝐴 ) = ( ( 𝑅 “ { 𝐵 } ) ∩ 𝐴 ) ) |
7 |
|
imass2 |
⊢ ( { 𝐵 } ⊆ 𝐴 → ( 𝑅 “ { 𝐵 } ) ⊆ ( 𝑅 “ 𝐴 ) ) |
8 |
2 7
|
syl |
⊢ ( ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( 𝑅 “ { 𝐵 } ) ⊆ ( 𝑅 “ 𝐴 ) ) |
9 |
|
simpl |
⊢ ( ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ) |
10 |
8 9
|
sstrd |
⊢ ( ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( 𝑅 “ { 𝐵 } ) ⊆ 𝐴 ) |
11 |
|
df-ss |
⊢ ( ( 𝑅 “ { 𝐵 } ) ⊆ 𝐴 ↔ ( ( 𝑅 “ { 𝐵 } ) ∩ 𝐴 ) = ( 𝑅 “ { 𝐵 } ) ) |
12 |
10 11
|
sylib |
⊢ ( ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( ( 𝑅 “ { 𝐵 } ) ∩ 𝐴 ) = ( 𝑅 “ { 𝐵 } ) ) |
13 |
6 12
|
eqtr2d |
⊢ ( ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( 𝑅 “ { 𝐵 } ) = ( ( 𝑅 “ ( { 𝐵 } ∩ 𝐴 ) ) ∩ 𝐴 ) ) |
14 |
|
imainrect |
⊢ ( ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) “ { 𝐵 } ) = ( ( 𝑅 “ ( { 𝐵 } ∩ 𝐴 ) ) ∩ 𝐴 ) |
15 |
13 14
|
eqtr4di |
⊢ ( ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( 𝑅 “ { 𝐵 } ) = ( ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) “ { 𝐵 } ) ) |
16 |
|
df-ec |
⊢ [ 𝐵 ] 𝑅 = ( 𝑅 “ { 𝐵 } ) |
17 |
|
df-ec |
⊢ [ 𝐵 ] ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) = ( ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) “ { 𝐵 } ) |
18 |
15 16 17
|
3eqtr4g |
⊢ ( ( ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ∧ 𝐵 ∈ 𝐴 ) → [ 𝐵 ] 𝑅 = [ 𝐵 ] ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) ) |