Metamath Proof Explorer


Theorem relcnveq3

Description: Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009)

Ref Expression
Assertion relcnveq3 RelRR=R-1xyxRyyRx

Proof

Step Hyp Ref Expression
1 eqss R=R-1RR-1R-1R
2 cnvsym R-1RxyxRyyRx
3 2 biimpi R-1RxyxRyyRx
4 3 a1d R-1RRelRxyxRyyRx
5 4 adantl RR-1R-1RRelRxyxRyyRx
6 5 com12 RelRRR-1R-1RxyxRyyRx
7 dfrel2 RelRR-1-1=R
8 cnvss R-1RR-1-1R-1
9 sseq1 R-1-1=RR-1-1R-1RR-1
10 8 9 syl5ibcom R-1RR-1-1=RRR-1
11 2 10 sylbir xyxRyyRxR-1-1=RRR-1
12 11 com12 R-1-1=RxyxRyyRxRR-1
13 7 12 sylbi RelRxyxRyyRxRR-1
14 2 biimpri xyxRyyRxR-1R
15 13 14 jca2 RelRxyxRyyRxRR-1R-1R
16 6 15 impbid RelRRR-1R-1RxyxRyyRx
17 1 16 bitrid RelRR=R-1xyxRyyRx