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Definition df-cnv 5319
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 ∈ V and 𝐵 ∈ V then (𝐴𝑅𝐵𝐵𝑅𝐴), as proven in brcnv 5506 (see df-br 4845 and df-rel 5318 for more on relations). For example, {⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩} (ex-cnv 27621). 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 "to the minus one." "Converse" is Quine's terminology; some authors call it "inverse," especially when the argument is a function. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
df-cnv 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
Distinct variable group:   𝑥,𝑦,𝐴

Detailed syntax breakdown of Definition df-cnv
StepHypRef Expression
1 cA . . 3 class 𝐴
21ccnv 5310 . 2 class 𝐴
3 vy . . . . 5 setvar 𝑦
43cv 1636 . . . 4 class 𝑦
5 vx . . . . 5 setvar 𝑥
65cv 1636 . . . 4 class 𝑥
74, 6, 1wbr 4844 . . 3 wff 𝑦𝐴𝑥
87, 5, 3copab 4906 . 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
92, 8wceq 1637 1 wff 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
Colors of variables: wff setvar class
This definition is referenced by:  cnvss  5496  elcnv  5500  nfcnv  5502  opelcnvg  5503  csbcnv  5507  csbcnvgALT  5508  cnvco  5509  relcnv  5713  cnv0  5746  cnvi  5748  cnvun  5749  cnvcnv3  5793
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