Step |
Hyp |
Ref |
Expression |
1 |
|
imaco |
⊢ ( ( 𝐹 ∘ 𝐺 ) “ { 𝑋 } ) = ( 𝐹 “ ( 𝐺 “ { 𝑋 } ) ) |
2 |
|
dfatsnafv2 |
⊢ ( 𝐺 defAt 𝑋 → { ( 𝐺 '''' 𝑋 ) } = ( 𝐺 “ { 𝑋 } ) ) |
3 |
2
|
adantr |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → { ( 𝐺 '''' 𝑋 ) } = ( 𝐺 “ { 𝑋 } ) ) |
4 |
3
|
imaeq2d |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( 𝐹 “ { ( 𝐺 '''' 𝑋 ) } ) = ( 𝐹 “ ( 𝐺 “ { 𝑋 } ) ) ) |
5 |
1 4
|
eqtr4id |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( ( 𝐹 ∘ 𝐺 ) “ { 𝑋 } ) = ( 𝐹 “ { ( 𝐺 '''' 𝑋 ) } ) ) |
6 |
5
|
eleq2d |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( 𝑥 ∈ ( ( 𝐹 ∘ 𝐺 ) “ { 𝑋 } ) ↔ 𝑥 ∈ ( 𝐹 “ { ( 𝐺 '''' 𝑋 ) } ) ) ) |
7 |
6
|
iotabidv |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( ℩ 𝑥 𝑥 ∈ ( ( 𝐹 ∘ 𝐺 ) “ { 𝑋 } ) ) = ( ℩ 𝑥 𝑥 ∈ ( 𝐹 “ { ( 𝐺 '''' 𝑋 ) } ) ) ) |
8 |
|
dfatco |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( 𝐹 ∘ 𝐺 ) defAt 𝑋 ) |
9 |
|
dfafv23 |
⊢ ( ( 𝐹 ∘ 𝐺 ) defAt 𝑋 → ( ( 𝐹 ∘ 𝐺 ) '''' 𝑋 ) = ( ℩ 𝑥 𝑥 ∈ ( ( 𝐹 ∘ 𝐺 ) “ { 𝑋 } ) ) ) |
10 |
8 9
|
syl |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( ( 𝐹 ∘ 𝐺 ) '''' 𝑋 ) = ( ℩ 𝑥 𝑥 ∈ ( ( 𝐹 ∘ 𝐺 ) “ { 𝑋 } ) ) ) |
11 |
|
dfafv23 |
⊢ ( 𝐹 defAt ( 𝐺 '''' 𝑋 ) → ( 𝐹 '''' ( 𝐺 '''' 𝑋 ) ) = ( ℩ 𝑥 𝑥 ∈ ( 𝐹 “ { ( 𝐺 '''' 𝑋 ) } ) ) ) |
12 |
11
|
adantl |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( 𝐹 '''' ( 𝐺 '''' 𝑋 ) ) = ( ℩ 𝑥 𝑥 ∈ ( 𝐹 “ { ( 𝐺 '''' 𝑋 ) } ) ) ) |
13 |
7 10 12
|
3eqtr4d |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( ( 𝐹 ∘ 𝐺 ) '''' 𝑋 ) = ( 𝐹 '''' ( 𝐺 '''' 𝑋 ) ) ) |