Step |
Hyp |
Ref |
Expression |
1 |
|
dfatcolem |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ∃! 𝑦 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) |
2 |
|
euex |
⊢ ( ∃! 𝑦 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 → ∃ 𝑦 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) |
3 |
1 2
|
syl |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ∃ 𝑦 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) |
4 |
|
df-dm |
⊢ dom ( 𝐹 ∘ 𝐺 ) = { 𝑥 ∣ ∃ 𝑦 𝑥 ( 𝐹 ∘ 𝐺 ) 𝑦 } |
5 |
4
|
eleq2i |
⊢ ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ↔ 𝑋 ∈ { 𝑥 ∣ ∃ 𝑦 𝑥 ( 𝐹 ∘ 𝐺 ) 𝑦 } ) |
6 |
|
df-dfat |
⊢ ( 𝐺 defAt 𝑋 ↔ ( 𝑋 ∈ dom 𝐺 ∧ Fun ( 𝐺 ↾ { 𝑋 } ) ) ) |
7 |
6
|
simplbi |
⊢ ( 𝐺 defAt 𝑋 → 𝑋 ∈ dom 𝐺 ) |
8 |
7
|
adantr |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → 𝑋 ∈ dom 𝐺 ) |
9 |
|
breq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ( 𝐹 ∘ 𝐺 ) 𝑦 ↔ 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) ) |
10 |
9
|
exbidv |
⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑦 𝑥 ( 𝐹 ∘ 𝐺 ) 𝑦 ↔ ∃ 𝑦 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) ) |
11 |
10
|
elabg |
⊢ ( 𝑋 ∈ dom 𝐺 → ( 𝑋 ∈ { 𝑥 ∣ ∃ 𝑦 𝑥 ( 𝐹 ∘ 𝐺 ) 𝑦 } ↔ ∃ 𝑦 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) ) |
12 |
8 11
|
syl |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( 𝑋 ∈ { 𝑥 ∣ ∃ 𝑦 𝑥 ( 𝐹 ∘ 𝐺 ) 𝑦 } ↔ ∃ 𝑦 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) ) |
13 |
5 12
|
syl5bb |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ↔ ∃ 𝑦 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) ) |
14 |
3 13
|
mpbird |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ) |
15 |
|
dfdfat2 |
⊢ ( ( 𝐹 ∘ 𝐺 ) defAt 𝑋 ↔ ( 𝑋 ∈ dom ( 𝐹 ∘ 𝐺 ) ∧ ∃! 𝑦 𝑋 ( 𝐹 ∘ 𝐺 ) 𝑦 ) ) |
16 |
14 1 15
|
sylanbrc |
⊢ ( ( 𝐺 defAt 𝑋 ∧ 𝐹 defAt ( 𝐺 '''' 𝑋 ) ) → ( 𝐹 ∘ 𝐺 ) defAt 𝑋 ) |