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 -1

Proof

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