Theorem f1o2d

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014)

	Ref	Expression
Hypotheses	f1od.1	|- F = ( x e. A |-> C )
	f1o2d.2	|- ( ( ph /\ x e. A ) -> C e. B )
	f1o2d.3	|- ( ( ph /\ y e. B ) -> D e. A )
	f1o2d.4	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )
Assertion	f1o2d	|- ( ph -> F : A -1-1-onto-> B )

Proof

Step	Hyp	Ref	Expression
1		f1od.1	|- F = ( x e. A |-> C )
2		f1o2d.2	|- ( ( ph /\ x e. A ) -> C e. B )
3		f1o2d.3	|- ( ( ph /\ y e. B ) -> D e. A )
4		f1o2d.4	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )
5	1 2 3 4	f1ocnv2d	|- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )
6	5	simpld	|- ( ph -> F : A -1-1-onto-> B )