Definition df-cnv 5122

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 (𝐴^◡𝑅𝐵 ↔ 𝐵𝑅𝐴), as proven in brcnv 5305 (see df-br 4654 and df-rel 5121 for more on relations). For example, ^◡{⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩} (ex-cnv 27294). 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

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	ccnv 5113	. 2 class ◡𝐴
3		vy	. . . . 5 setvar 𝑦
4	3	cv 1482	. . . 4 class 𝑦
5		vx	. . . . 5 setvar 𝑥
6	5	cv 1482	. . . 4 class 𝑥
7	4, 6, 1	wbr 4653	. . 3 wff 𝑦𝐴𝑥
8	7, 5, 3	copab 4712	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
9	2, 8	wceq 1483	1 wff ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}

This definition is referenced by:  cnvss  5294  cnvssOLD  5295  elcnv  5299  nfcnv  5301  opelcnvg  5302  csbcnv  5306  csbcnvgALT  5307  cnvco  5308  relcnv  5503  cnv0  5535  cnvi  5537  cnvun  5538  cnvcnv3  5582