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Definition df-cnv 5151
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 5337 (see df-br 4686 and df-rel 5150 for more on relations). For example, {⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩} (ex-cnv 27424). 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 5142 . 2 class 𝐴
3 vy . . . . 5 setvar 𝑦
43cv 1522 . . . 4 class 𝑦
5 vx . . . . 5 setvar 𝑥
65cv 1522 . . . 4 class 𝑥
74, 6, 1wbr 4685 . . 3 wff 𝑦𝐴𝑥
87, 5, 3copab 4745 . 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
92, 8wceq 1523 1 wff 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
Colors of variables: wff setvar class
This definition is referenced by:  cnvss  5327  elcnv  5331  nfcnv  5333  opelcnvg  5334  csbcnv  5338  csbcnvgALT  5339  cnvco  5340  relcnv  5538  cnv0  5570  cnvi  5572  cnvun  5573  cnvcnv3  5617
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