Definition df-cnv 5365

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 5552 (see df-br 4889 and df-rel 5364 for more on relations). For example, ^◡{⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩} (ex-cnv 27886). 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 5356	. 2 class ◡𝐴
3		vy	. . . . 5 setvar 𝑦
4	3	cv 1600	. . . 4 class 𝑦
5		vx	. . . . 5 setvar 𝑥
6	5	cv 1600	. . . 4 class 𝑥
7	4, 6, 1	wbr 4888	. . 3 wff 𝑦𝐴𝑥
8	7, 5, 3	copab 4950	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
9	2, 8	wceq 1601	1 wff ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}

This definition is referenced by:  cnvss  5542  elcnv  5546  nfcnv  5548  brcnvg  5549  csbcnv  5553  csbcnvgALT  5554  cnvco  5555  relcnv  5759  cnv0  5792  cnvi  5793  cnvun  5794  cnvcnv3  5838