Theorem sprsymrelen

Description: The class P of subsets of the set of pairs over a fixed set V and the class R of symmetric relations on the fixed set V are equinumerous. (Contributed by AV, 27-Nov-2021)

	Ref	Expression
Hypotheses	sprsymrelf.p	$$⊢P=𝒫\mathrm{Pairs}\left(V\right)$$
	sprsymrelf.r	$$⊢R=\left\{r\in 𝒫\left(V\times V\right)|\forall x\in V\forall y\in V\left(xry\leftrightarrow yrx\right)\right\}$$
Assertion	sprsymrelen	$$⊢V\in W\to P\approx R$$

Proof

Step	Hyp	Ref	Expression
1		sprsymrelf.p	$$⊢P=𝒫\mathrm{Pairs}\left(V\right)$$
2		sprsymrelf.r	$$⊢R=\left\{r\in 𝒫\left(V\times V\right)|\forall x\in V\forall y\in V\left(xry\leftrightarrow yrx\right)\right\}$$
3	1 2	sprbisymrel	$$⊢V\in W\to \exists ff:P\underset{1-1\; onto}{⟶}R$$
4		bren	$$⊢P\approx R\leftrightarrow \exists ff:P\underset{1-1\; onto}{⟶}R$$
5	3 4	sylibr	$$⊢V\in W\to P\approx R$$