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Metamath Proof Explorer

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 ) ) )