Theorem dff1o6

Description: A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008)

	Ref	Expression
Assertion	dff1o6	|- ( F : A -1-1-onto-> B <-> ( F Fn A /\ ran F = B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-f1o	|- ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2		dff13	|- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
3		df-fo	|- ( F : A -onto-> B <-> ( F Fn A /\ ran F = B ) )
4	2 3	anbi12i	|- ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) /\ ( F Fn A /\ ran F = B ) ) )
5		df-3an	|- ( ( F Fn A /\ ran F = B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) <-> ( ( F Fn A /\ ran F = B ) /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
6		eqimss	|- ( ran F = B -> ran F C_ B )
7	6	anim2i	|- ( ( F Fn A /\ ran F = B ) -> ( F Fn A /\ ran F C_ B ) )
8		df-f	|- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
9	7 8	sylibr	|- ( ( F Fn A /\ ran F = B ) -> F : A --> B )
10	9	pm4.71ri	|- ( ( F Fn A /\ ran F = B ) <-> ( F : A --> B /\ ( F Fn A /\ ran F = B ) ) )
11	10	anbi1i	|- ( ( ( F Fn A /\ ran F = B ) /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) <-> ( ( F : A --> B /\ ( F Fn A /\ ran F = B ) ) /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
12		an32	|- ( ( ( F : A --> B /\ ( F Fn A /\ ran F = B ) ) /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) <-> ( ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) /\ ( F Fn A /\ ran F = B ) ) )
13	5 11 12	3bitrri	|- ( ( ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) /\ ( F Fn A /\ ran F = B ) ) <-> ( F Fn A /\ ran F = B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
14	1 4 13	3bitri	|- ( F : A -1-1-onto-> B <-> ( F Fn A /\ ran F = B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )