Definition df-cnv 5563

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 5753 (see df-br 5067 and df-rel 5562 for more on relations). For example, ^◡{⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩} (ex-cnv 28216). 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 "minus one". The term "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 5554	. 2 class ◡𝐴
3		vy	. . . . 5 setvar 𝑦
4	3	cv 1536	. . . 4 class 𝑦
5		vx	. . . . 5 setvar 𝑥
6	5	cv 1536	. . . 4 class 𝑥
7	4, 6, 1	wbr 5066	. . 3 wff 𝑦𝐴𝑥
8	7, 5, 3	copab 5128	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
9	2, 8	wceq 1537	1 wff ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}

This definition is referenced by:  cnvss  5743  elcnv  5747  nfcnv  5749  brcnvg  5750  csbcnv  5754  csbcnvgALT  5755  cnvco  5756  relcnv  5967  cnv0  5999  cnvi  6000  cnvun  6001  cnvcnv3  6045