Step |
Hyp |
Ref |
Expression |
1 |
|
df-funpart |
⊢ Funpart 𝐹 = ( 𝐹 ↾ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) ) |
2 |
1
|
fveq1i |
⊢ ( Funpart 𝐹 ‘ 𝐴 ) = ( ( 𝐹 ↾ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) ) ‘ 𝐴 ) |
3 |
|
fvres |
⊢ ( 𝐴 ∈ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) → ( ( 𝐹 ↾ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) |
4 |
|
nfvres |
⊢ ( ¬ 𝐴 ∈ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) → ( ( 𝐹 ↾ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) ) ‘ 𝐴 ) = ∅ ) |
5 |
|
funpartlem |
⊢ ( 𝐴 ∈ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) ↔ ∃ 𝑥 ( 𝐹 “ { 𝐴 } ) = { 𝑥 } ) |
6 |
|
eusn |
⊢ ( ∃! 𝑥 𝑥 ∈ ( 𝐹 “ { 𝐴 } ) ↔ ∃ 𝑥 ( 𝐹 “ { 𝐴 } ) = { 𝑥 } ) |
7 |
5 6
|
bitr4i |
⊢ ( 𝐴 ∈ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) ↔ ∃! 𝑥 𝑥 ∈ ( 𝐹 “ { 𝐴 } ) ) |
8 |
|
elimasng |
⊢ ( ( 𝐴 ∈ V ∧ 𝑥 ∈ V ) → ( 𝑥 ∈ ( 𝐹 “ { 𝐴 } ) ↔ 〈 𝐴 , 𝑥 〉 ∈ 𝐹 ) ) |
9 |
8
|
elvd |
⊢ ( 𝐴 ∈ V → ( 𝑥 ∈ ( 𝐹 “ { 𝐴 } ) ↔ 〈 𝐴 , 𝑥 〉 ∈ 𝐹 ) ) |
10 |
|
df-br |
⊢ ( 𝐴 𝐹 𝑥 ↔ 〈 𝐴 , 𝑥 〉 ∈ 𝐹 ) |
11 |
9 10
|
bitr4di |
⊢ ( 𝐴 ∈ V → ( 𝑥 ∈ ( 𝐹 “ { 𝐴 } ) ↔ 𝐴 𝐹 𝑥 ) ) |
12 |
11
|
eubidv |
⊢ ( 𝐴 ∈ V → ( ∃! 𝑥 𝑥 ∈ ( 𝐹 “ { 𝐴 } ) ↔ ∃! 𝑥 𝐴 𝐹 𝑥 ) ) |
13 |
7 12
|
syl5bb |
⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) ↔ ∃! 𝑥 𝐴 𝐹 𝑥 ) ) |
14 |
13
|
notbid |
⊢ ( 𝐴 ∈ V → ( ¬ 𝐴 ∈ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) ↔ ¬ ∃! 𝑥 𝐴 𝐹 𝑥 ) ) |
15 |
|
tz6.12-2 |
⊢ ( ¬ ∃! 𝑥 𝐴 𝐹 𝑥 → ( 𝐹 ‘ 𝐴 ) = ∅ ) |
16 |
14 15
|
syl6bi |
⊢ ( 𝐴 ∈ V → ( ¬ 𝐴 ∈ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) → ( 𝐹 ‘ 𝐴 ) = ∅ ) ) |
17 |
|
fvprc |
⊢ ( ¬ 𝐴 ∈ V → ( 𝐹 ‘ 𝐴 ) = ∅ ) |
18 |
17
|
a1d |
⊢ ( ¬ 𝐴 ∈ V → ( ¬ 𝐴 ∈ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) → ( 𝐹 ‘ 𝐴 ) = ∅ ) ) |
19 |
16 18
|
pm2.61i |
⊢ ( ¬ 𝐴 ∈ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) → ( 𝐹 ‘ 𝐴 ) = ∅ ) |
20 |
4 19
|
eqtr4d |
⊢ ( ¬ 𝐴 ∈ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) → ( ( 𝐹 ↾ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) |
21 |
3 20
|
pm2.61i |
⊢ ( ( 𝐹 ↾ dom ( ( Image 𝐹 ∘ Singleton ) ∩ ( V × Singletons ) ) ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) |
22 |
2 21
|
eqtri |
⊢ ( Funpart 𝐹 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) |