Definition df-cnv 4371

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 4536 (see df-br 3786 and df-rel 4370 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. 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 4362	. 2 class ◡𝐴
3		vy	. . . . 5 setvar 𝑦
4	3	cv 1283	. . . 4 class 𝑦
5		vx	. . . . 5 setvar 𝑥
6	5	cv 1283	. . . 4 class 𝑥
7	4, 6, 1	wbr 3785	. . 3 wff 𝑦𝐴𝑥
8	7, 5, 3	copab 3838	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
9	2, 8	wceq 1284	1 wff ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}

This definition is referenced by:  cnvss  4526  elcnv  4530  nfcnv  4532  opelcnvg  4533  csbcnvg  4537  cnvco  4538  relcnv  4723  cnvi  4748  cnvun  4749  cnvin  4751  cnvcnv3  4790