Step |
Hyp |
Ref |
Expression |
1 |
|
funimage |
⊢ Fun Image 𝑅 |
2 |
|
vex |
⊢ 𝑦 ∈ V |
3 |
|
vex |
⊢ 𝑥 ∈ V |
4 |
2 3
|
brimage |
⊢ ( 𝑦 Image 𝑅 𝑥 ↔ 𝑥 = ( 𝑅 “ 𝑦 ) ) |
5 |
|
eqvisset |
⊢ ( 𝑥 = ( 𝑅 “ 𝑦 ) → ( 𝑅 “ 𝑦 ) ∈ V ) |
6 |
4 5
|
sylbi |
⊢ ( 𝑦 Image 𝑅 𝑥 → ( 𝑅 “ 𝑦 ) ∈ V ) |
7 |
6
|
exlimiv |
⊢ ( ∃ 𝑥 𝑦 Image 𝑅 𝑥 → ( 𝑅 “ 𝑦 ) ∈ V ) |
8 |
|
eqid |
⊢ ( 𝑅 “ 𝑦 ) = ( 𝑅 “ 𝑦 ) |
9 |
|
brimageg |
⊢ ( ( 𝑦 ∈ V ∧ ( 𝑅 “ 𝑦 ) ∈ V ) → ( 𝑦 Image 𝑅 ( 𝑅 “ 𝑦 ) ↔ ( 𝑅 “ 𝑦 ) = ( 𝑅 “ 𝑦 ) ) ) |
10 |
2 9
|
mpan |
⊢ ( ( 𝑅 “ 𝑦 ) ∈ V → ( 𝑦 Image 𝑅 ( 𝑅 “ 𝑦 ) ↔ ( 𝑅 “ 𝑦 ) = ( 𝑅 “ 𝑦 ) ) ) |
11 |
8 10
|
mpbiri |
⊢ ( ( 𝑅 “ 𝑦 ) ∈ V → 𝑦 Image 𝑅 ( 𝑅 “ 𝑦 ) ) |
12 |
|
breq2 |
⊢ ( 𝑥 = ( 𝑅 “ 𝑦 ) → ( 𝑦 Image 𝑅 𝑥 ↔ 𝑦 Image 𝑅 ( 𝑅 “ 𝑦 ) ) ) |
13 |
12
|
spcegv |
⊢ ( ( 𝑅 “ 𝑦 ) ∈ V → ( 𝑦 Image 𝑅 ( 𝑅 “ 𝑦 ) → ∃ 𝑥 𝑦 Image 𝑅 𝑥 ) ) |
14 |
11 13
|
mpd |
⊢ ( ( 𝑅 “ 𝑦 ) ∈ V → ∃ 𝑥 𝑦 Image 𝑅 𝑥 ) |
15 |
7 14
|
impbii |
⊢ ( ∃ 𝑥 𝑦 Image 𝑅 𝑥 ↔ ( 𝑅 “ 𝑦 ) ∈ V ) |
16 |
2
|
eldm |
⊢ ( 𝑦 ∈ dom Image 𝑅 ↔ ∃ 𝑥 𝑦 Image 𝑅 𝑥 ) |
17 |
|
imaeq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑅 “ 𝑥 ) = ( 𝑅 “ 𝑦 ) ) |
18 |
17
|
eleq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑅 “ 𝑥 ) ∈ V ↔ ( 𝑅 “ 𝑦 ) ∈ V ) ) |
19 |
2 18
|
elab |
⊢ ( 𝑦 ∈ { 𝑥 ∣ ( 𝑅 “ 𝑥 ) ∈ V } ↔ ( 𝑅 “ 𝑦 ) ∈ V ) |
20 |
15 16 19
|
3bitr4i |
⊢ ( 𝑦 ∈ dom Image 𝑅 ↔ 𝑦 ∈ { 𝑥 ∣ ( 𝑅 “ 𝑥 ) ∈ V } ) |
21 |
20
|
eqriv |
⊢ dom Image 𝑅 = { 𝑥 ∣ ( 𝑅 “ 𝑥 ) ∈ V } |
22 |
|
df-fn |
⊢ ( Image 𝑅 Fn { 𝑥 ∣ ( 𝑅 “ 𝑥 ) ∈ V } ↔ ( Fun Image 𝑅 ∧ dom Image 𝑅 = { 𝑥 ∣ ( 𝑅 “ 𝑥 ) ∈ V } ) ) |
23 |
1 21 22
|
mpbir2an |
⊢ Image 𝑅 Fn { 𝑥 ∣ ( 𝑅 “ 𝑥 ) ∈ V } |