Step |
Hyp |
Ref |
Expression |
1 |
|
eqcom |
⊢ ( 𝑦 = ( 𝐹 '''' 𝐴 ) ↔ ( 𝐹 '''' 𝐴 ) = 𝑦 ) |
2 |
|
dfatbrafv2b |
⊢ ( ( 𝐹 defAt 𝐴 ∧ 𝑦 ∈ V ) → ( ( 𝐹 '''' 𝐴 ) = 𝑦 ↔ 𝐴 𝐹 𝑦 ) ) |
3 |
2
|
elvd |
⊢ ( 𝐹 defAt 𝐴 → ( ( 𝐹 '''' 𝐴 ) = 𝑦 ↔ 𝐴 𝐹 𝑦 ) ) |
4 |
1 3
|
syl5bb |
⊢ ( 𝐹 defAt 𝐴 → ( 𝑦 = ( 𝐹 '''' 𝐴 ) ↔ 𝐴 𝐹 𝑦 ) ) |
5 |
4
|
abbidv |
⊢ ( 𝐹 defAt 𝐴 → { 𝑦 ∣ 𝑦 = ( 𝐹 '''' 𝐴 ) } = { 𝑦 ∣ 𝐴 𝐹 𝑦 } ) |
6 |
|
df-sn |
⊢ { ( 𝐹 '''' 𝐴 ) } = { 𝑦 ∣ 𝑦 = ( 𝐹 '''' 𝐴 ) } |
7 |
6
|
a1i |
⊢ ( 𝐹 defAt 𝐴 → { ( 𝐹 '''' 𝐴 ) } = { 𝑦 ∣ 𝑦 = ( 𝐹 '''' 𝐴 ) } ) |
8 |
|
dfdfat2 |
⊢ ( 𝐹 defAt 𝐴 ↔ ( 𝐴 ∈ dom 𝐹 ∧ ∃! 𝑥 𝐴 𝐹 𝑥 ) ) |
9 |
|
imasng |
⊢ ( 𝐴 ∈ dom 𝐹 → ( 𝐹 “ { 𝐴 } ) = { 𝑦 ∣ 𝐴 𝐹 𝑦 } ) |
10 |
9
|
adantr |
⊢ ( ( 𝐴 ∈ dom 𝐹 ∧ ∃! 𝑥 𝐴 𝐹 𝑥 ) → ( 𝐹 “ { 𝐴 } ) = { 𝑦 ∣ 𝐴 𝐹 𝑦 } ) |
11 |
8 10
|
sylbi |
⊢ ( 𝐹 defAt 𝐴 → ( 𝐹 “ { 𝐴 } ) = { 𝑦 ∣ 𝐴 𝐹 𝑦 } ) |
12 |
5 7 11
|
3eqtr4d |
⊢ ( 𝐹 defAt 𝐴 → { ( 𝐹 '''' 𝐴 ) } = ( 𝐹 “ { 𝐴 } ) ) |