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. "Converse" is Quine's terminology. Some authors use a "minus one" exponent and call it "inverse", especially when the argument is a function, although this is not in general a genuine inverse. (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 }