Definition df-cnv 4782

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 4963 (see df-br 4131 and df-rel 4781 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

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	ccnv 4773	. 2 class ◡𝐴
3		vy	. . . . 5 setvar 𝑦
4	3	cv 1401	. . . 4 class 𝑦
5		vx	. . . . 5 setvar 𝑥
6	5	cv 1401	. . . 4 class 𝑥
7	4, 6, 1	wbr 4130	. . 3 wff 𝑦𝐴𝑥
8	7, 5, 3	copab 4191	. 2 class {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
9	2, 8	wceq 1402	1 wff ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}

This definition is used by:  cnvss  4953  elcnv  4957  nfcnv  4959  opelcnvg  4960  csbcnvg  4964  cnvco  4965  relcnv  5165  cnvi  5192  cnvun  5193  cnvin  5195  cnvcnv3  5237