Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-cnv | Structured version Visualization version GIF version |
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 5227 (see df-br 4584 and df-rel 5045 for more on relations). For example, ◡{〈2, 6〉, 〈3, 9〉} = {〈6, 2〉, 〈9, 3〉} (ex-cnv 26686). 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.) |
Ref | Expression |
---|---|
df-cnv | ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | 1 | ccnv 5037 | . 2 class ◡𝐴 |
3 | vy | . . . . 5 setvar 𝑦 | |
4 | 3 | cv 1474 | . . . 4 class 𝑦 |
5 | vx | . . . . 5 setvar 𝑥 | |
6 | 5 | cv 1474 | . . . 4 class 𝑥 |
7 | 4, 6, 1 | wbr 4583 | . . 3 wff 𝑦𝐴𝑥 |
8 | 7, 5, 3 | copab 4642 | . 2 class {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} |
9 | 2, 8 | wceq 1475 | 1 wff ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} |
Colors of variables: wff setvar class |
This definition is referenced by: cnvss 5216 cnvssOLD 5217 elcnv 5221 nfcnv 5223 opelcnvg 5224 csbcnv 5228 csbcnvgALT 5229 cnvco 5230 relcnv 5422 cnv0 5454 cnvi 5456 cnvun 5457 cnvcnv3 5501 |
Copyright terms: Public domain | W3C validator |