# Metamath Proof Explorer

## Definition df-cnv

Description: Define the converse of a class. Definition 9.12 of Quine p. 64. The converse of a binary relation swaps its arguments, i.e., if A e.V and B e. V then ( A`' R B <-> B R A ) , as proven in brcnv (see df-br and df-rel for more on relations). For example, ``' { <. 2 , 6 >. , <. 3 , 9 >. } = { <. 6 , 2 >. , <. 9 , 3 >. } ` ( ex-cnv ). We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. Many authors use the postfix superscript "minus one". The term "converse" is Quine's terminology; some authors call it "inverse", especially when the argument is a function. (Contributed by NM, 4-Jul-1994)

Ref Expression
Assertion df-cnv
`|- `' A = { <. x , y >. | y A x }`

### Detailed syntax breakdown

Step Hyp Ref Expression
0 cA
` |-  A`
1 0 ccnv
` |-  `' A`
2 vx
` |-  x`
3 vy
` |-  y`
4 3 cv
` |-  y`
5 2 cv
` |-  x`
6 4 5 0 wbr
` |-  y A x`
7 6 2 3 copab
` |-  { <. x , y >. | y A x }`
8 1 7 wceq
` |-  `' A = { <. x , y >. | y A x }`