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 }