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Mirrors > Home > ILE Home > Th. List > df-cnv | 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 4692 (see df-br 3900 and df-rel 4516 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.) |
Ref | Expression |
---|---|
df-cnv | ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | 1 | ccnv 4508 | . 2 class ◡𝐴 |
3 | vy | . . . . 5 setvar 𝑦 | |
4 | 3 | cv 1315 | . . . 4 class 𝑦 |
5 | vx | . . . . 5 setvar 𝑥 | |
6 | 5 | cv 1315 | . . . 4 class 𝑥 |
7 | 4, 6, 1 | wbr 3899 | . . 3 wff 𝑦𝐴𝑥 |
8 | 7, 5, 3 | copab 3958 | . 2 class {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} |
9 | 2, 8 | wceq 1316 | 1 wff ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} |
Colors of variables: wff set class |
This definition is referenced by: cnvss 4682 elcnv 4686 nfcnv 4688 opelcnvg 4689 csbcnvg 4693 cnvco 4694 relcnv 4887 cnvi 4913 cnvun 4914 cnvin 4916 cnvcnv3 4958 |
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