Metamath Proof Explorer


Theorem idsset

Description: _I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012)

Ref Expression
Assertion idsset I = ( SSet SSet )

Proof

Step Hyp Ref Expression
1 reli Rel I
2 relsset Rel SSet
3 relin1 ( Rel SSet → Rel ( SSet SSet ) )
4 2 3 ax-mp Rel ( SSet SSet )
5 eqss ( 𝑦 = 𝑧 ↔ ( 𝑦𝑧𝑧𝑦 ) )
6 vex 𝑧 ∈ V
7 6 ideq ( 𝑦 I 𝑧𝑦 = 𝑧 )
8 brin ( 𝑦 ( SSet SSet ) 𝑧 ↔ ( 𝑦 SSet 𝑧𝑦 SSet 𝑧 ) )
9 6 brsset ( 𝑦 SSet 𝑧𝑦𝑧 )
10 vex 𝑦 ∈ V
11 10 6 brcnv ( 𝑦 SSet 𝑧𝑧 SSet 𝑦 )
12 10 brsset ( 𝑧 SSet 𝑦𝑧𝑦 )
13 11 12 bitri ( 𝑦 SSet 𝑧𝑧𝑦 )
14 9 13 anbi12i ( ( 𝑦 SSet 𝑧𝑦 SSet 𝑧 ) ↔ ( 𝑦𝑧𝑧𝑦 ) )
15 8 14 bitri ( 𝑦 ( SSet SSet ) 𝑧 ↔ ( 𝑦𝑧𝑧𝑦 ) )
16 5 7 15 3bitr4i ( 𝑦 I 𝑧𝑦 ( SSet SSet ) 𝑧 )
17 1 4 16 eqbrriv I = ( SSet SSet )