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Definition df-cnv 5266
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 5452 (see df-br 4797 and df-rel 5265 for more on relations). For example, {⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩} (ex-cnv 27597). 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 5257 . 2 class 𝐴
3 vy . . . . 5 setvar 𝑦
43cv 1623 . . . 4 class 𝑦
5 vx . . . . 5 setvar 𝑥
65cv 1623 . . . 4 class 𝑥
74, 6, 1wbr 4796 . . 3 wff 𝑦𝐴𝑥
87, 5, 3copab 4856 . 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
92, 8wceq 1624 1 wff 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
Colors of variables: wff setvar class
This definition is referenced by:  cnvss  5442  elcnv  5446  nfcnv  5448  opelcnvg  5449  csbcnv  5453  csbcnvgALT  5454  cnvco  5455  relcnv  5653  cnv0  5685  cnvi  5687  cnvun  5688  cnvcnv3  5732
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