Theorem en3d

Description: Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004) (Revised by Mario Carneiro, 12-May-2014) (Revised by AV, 4-Aug-2024)

	Ref	Expression
Hypotheses	en3d.1	|- ( ph -> A e. V )
	en3d.2	|- ( ph -> B e. W )
	en3d.3	|- ( ph -> ( x e. A -> C e. B ) )
	en3d.4	|- ( ph -> ( y e. B -> D e. A ) )
	en3d.5	|- ( ph -> ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) ) )
Assertion	en3d	|- ( ph -> A ~~ B )

Proof

Step	Hyp	Ref	Expression
1		en3d.1	|- ( ph -> A e. V )
2		en3d.2	|- ( ph -> B e. W )
3		en3d.3	|- ( ph -> ( x e. A -> C e. B ) )
4		en3d.4	|- ( ph -> ( y e. B -> D e. A ) )
5		en3d.5	|- ( ph -> ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) ) )
6		eqid	|- ( x e. A |-> C ) = ( x e. A |-> C )
7	3	imp	|- ( ( ph /\ x e. A ) -> C e. B )
8	4	imp	|- ( ( ph /\ y e. B ) -> D e. A )
9	5	imp	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )
10	6 7 8 9	f1o2d	|- ( ph -> ( x e. A |-> C ) : A -1-1-onto-> B )
11		f1oen2g	|- ( ( A e. V /\ B e. W /\ ( x e. A |-> C ) : A -1-1-onto-> B ) -> A ~~ B )
12	1 2 10 11	syl3anc	|- ( ph -> A ~~ B )