Metamath Proof Explorer


Theorem qsid

Description: A set is equal to its quotient set modulo the converse membership relation. (Note: the converse membership relation is not an equivalence relation.) (Contributed by NM, 13-Aug-1995) (Revised by Mario Carneiro, 9-Jul-2014)

Ref Expression
Assertion qsid ( 𝐴 / E ) = 𝐴

Proof

Step Hyp Ref Expression
1 vex 𝑥 ∈ V
2 1 ecid [ 𝑥 ] E = 𝑥
3 2 eqeq2i ( 𝑦 = [ 𝑥 ] E ↔ 𝑦 = 𝑥 )
4 equcom ( 𝑦 = 𝑥𝑥 = 𝑦 )
5 3 4 bitri ( 𝑦 = [ 𝑥 ] E ↔ 𝑥 = 𝑦 )
6 5 rexbii ( ∃ 𝑥𝐴 𝑦 = [ 𝑥 ] E ↔ ∃ 𝑥𝐴 𝑥 = 𝑦 )
7 vex 𝑦 ∈ V
8 7 elqs ( 𝑦 ∈ ( 𝐴 / E ) ↔ ∃ 𝑥𝐴 𝑦 = [ 𝑥 ] E )
9 risset ( 𝑦𝐴 ↔ ∃ 𝑥𝐴 𝑥 = 𝑦 )
10 6 8 9 3bitr4i ( 𝑦 ∈ ( 𝐴 / E ) ↔ 𝑦𝐴 )
11 10 eqriv ( 𝐴 / E ) = 𝐴