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 ( Pairs ` V )
	sprsymrelf.r	|- R = { r e. ~P ( V X. V ) | A. x e. V A. y e. V ( x r y <-> y r x ) }
Assertion	sprsymrelen	|- ( V e. W -> P ~~ R )

Proof

Step	Hyp	Ref	Expression
1		sprsymrelf.p	|- P = ~P ( Pairs ` V )
2		sprsymrelf.r	|- R = { r e. ~P ( V X. V ) | A. x e. V A. y e. V ( x r y <-> y r x ) }
3	1 2	sprbisymrel	|- ( V e. W -> E. f f : P -1-1-onto-> R )
4		bren	|- ( P ~~ R <-> E. f f : P -1-1-onto-> R )
5	3 4	sylibr	|- ( V e. W -> P ~~ R )