Metamath Proof Explorer
Description: Any set is equal to its preimage under the converse membership relation.
(Contributed by Mario Carneiro, 9-Mar-2013)
|
|
Ref |
Expression |
|
Hypothesis |
epini.1 |
⊢ 𝐴 ∈ V |
|
Assertion |
epini |
⊢ ( ◡ E “ { 𝐴 } ) = 𝐴 |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
epini.1 |
⊢ 𝐴 ∈ V |
2 |
|
vex |
⊢ 𝑥 ∈ V |
3 |
2
|
eliniseg |
⊢ ( 𝐴 ∈ V → ( 𝑥 ∈ ( ◡ E “ { 𝐴 } ) ↔ 𝑥 E 𝐴 ) ) |
4 |
1 3
|
ax-mp |
⊢ ( 𝑥 ∈ ( ◡ E “ { 𝐴 } ) ↔ 𝑥 E 𝐴 ) |
5 |
1
|
epeli |
⊢ ( 𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴 ) |
6 |
4 5
|
bitri |
⊢ ( 𝑥 ∈ ( ◡ E “ { 𝐴 } ) ↔ 𝑥 ∈ 𝐴 ) |
7 |
6
|
eqriv |
⊢ ( ◡ E “ { 𝐴 } ) = 𝐴 |