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Definition df-cnv 4636
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 4812 (see df-br 4006 and df-rel 4635 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⟩ ∣ 𝑦𝐴𝑥}
Distinct variable group:   𝑥,𝑦,𝐴

Detailed syntax breakdown of Definition df-cnv
StepHypRef Expression
1 cA . . 3 class 𝐴
21ccnv 4627 . 2 class 𝐴
3 vy . . . . 5 setvar 𝑦
43cv 1352 . . . 4 class 𝑦
5 vx . . . . 5 setvar 𝑥
65cv 1352 . . . 4 class 𝑥
74, 6, 1wbr 4005 . . 3 wff 𝑦𝐴𝑥
87, 5, 3copab 4065 . 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
92, 8wceq 1353 1 wff 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
Colors of variables: wff set class
This definition is referenced by:  cnvss  4802  elcnv  4806  nfcnv  4808  opelcnvg  4809  csbcnvg  4813  cnvco  4814  relcnv  5008  cnvi  5035  cnvun  5036  cnvin  5038  cnvcnv3  5080
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