Theorem bren

Description: Equinumerosity relation. (Contributed by NM, 15-Jun-1998) Extract breng as an intermediate result. (Revised by BTernaryTau, 23-Sep-2024)

	Ref	Expression
Assertion	bren	|- ( A ~~ B <-> E. f f : A -1-1-onto-> B )

Proof

Step	Hyp	Ref	Expression
1		encv	|- ( A ~~ B -> ( A e. _V /\ B e. _V ) )
2		f1ofn	|- ( f : A -1-1-onto-> B -> f Fn A )
3		fndm	|- ( f Fn A -> dom f = A )
4		vex	|- f e. _V
5	4	dmex	|- dom f e. _V
6	3 5	eqeltrrdi	|- ( f Fn A -> A e. _V )
7	2 6	syl	|- ( f : A -1-1-onto-> B -> A e. _V )
8		f1ofo	|- ( f : A -1-1-onto-> B -> f : A -onto-> B )
9		forn	|- ( f : A -onto-> B -> ran f = B )
10	8 9	syl	|- ( f : A -1-1-onto-> B -> ran f = B )
11	4	rnex	|- ran f e. _V
12	10 11	eqeltrrdi	|- ( f : A -1-1-onto-> B -> B e. _V )
13	7 12	jca	|- ( f : A -1-1-onto-> B -> ( A e. _V /\ B e. _V ) )
14	13	exlimiv	|- ( E. f f : A -1-1-onto-> B -> ( A e. _V /\ B e. _V ) )
15		breng	|- ( ( A e. _V /\ B e. _V ) -> ( A ~~ B <-> E. f f : A -1-1-onto-> B ) )
16	1 14 15	pm5.21nii	|- ( A ~~ B <-> E. f f : A -1-1-onto-> B )