Definition df-cnv 4782

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 4963 (see df-br 4131 and df-rel 4781 for more on relations). For example, `' { <. 2 , 6 >. , <. 3 , 9 >. } = { <. 6 , 2 >. , <. 9 , 3 >. }.

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.)

Assertion

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

Distinct variable group:   x, y, A

Detailed syntax breakdown of Definition df-cnv

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2	1	ccnv 4773	. 2 class `' A
3		vy	. . . . 5 setvar y
4	3	cv 1401	. . . 4 class y
5		vx	. . . . 5 setvar x
6	5	cv 1401	. . . 4 class x
7	4, 6, 1	wbr 4130	. . 3 wff y A x
8	7, 5, 3	copab 4191	. 2 class { <. x , y >. | y A x }
9	2, 8	wceq 1402	1 wff `' A = { <. x , y >. | y A x }

This definition is used by:  cnvss  4953  elcnv  4957  nfcnv  4959  opelcnvg  4960  csbcnvg  4964  cnvco  4965  relcnv  5165  cnvi  5192  cnvun  5193  cnvin  5195  cnvcnv3  5237