Metamath Proof Explorer

Theorem dff1o2

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997) (Proof shortened by Andrew Salmon, 22-Oct-2011)

Ref Expression
Assertion dff1o2 
|- ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )

Proof

Step Hyp Ref Expression
1 df-f1o 
 |-  ( F : A -1-1-onto-> B <-> ( F : A -1-1-> B /\ F : A -onto-> B ) )
2 df-f1 
 |-  ( F : A -1-1-> B <-> ( F : A --> B /\ Fun `' F ) )
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 /\ Fun `' F ) /\ ( F Fn A /\ ran F = B ) ) )
5 anass 
 |-  ( ( ( F : A --> B /\ Fun `' F ) /\ ( F Fn A /\ ran F = B ) ) <-> ( F : A --> B /\ ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) ) )
6 3anan12 
 |-  ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) )
7 6 anbi1i 
 |-  ( ( ( F Fn A /\ Fun `' F /\ ran F = B ) /\ F : A --> B ) <-> ( ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) /\ F : A --> B ) )
8 eqimss 
 |-  ( ran F = B -> ran F C_ B )
9 df-f 
 |-  ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
10 9 biimpri 
 |-  ( ( F Fn A /\ ran F C_ B ) -> F : A --> B )
11 8 10 sylan2 
 |-  ( ( F Fn A /\ ran F = B ) -> F : A --> B )
12 11 3adant2 
 |-  ( ( F Fn A /\ Fun `' F /\ ran F = B ) -> F : A --> B )
13 12 pm4.71i 
 |-  ( ( F Fn A /\ Fun `' F /\ ran F = B ) <-> ( ( F Fn A /\ Fun `' F /\ ran F = B ) /\ F : A --> B ) )
14 ancom 
 |-  ( ( F : A --> B /\ ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) ) <-> ( ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) /\ F : A --> B ) )
15 7 13 14 3bitr4ri 
 |-  ( ( F : A --> B /\ ( Fun `' F /\ ( F Fn A /\ ran F = B ) ) ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
16 5 15 bitri 
 |-  ( ( ( F : A --> B /\ Fun `' F ) /\ ( F Fn A /\ ran F = B ) ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
17 4 16 bitri 
 |-  ( ( F : A -1-1-> B /\ F : A -onto-> B ) <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )
18 1 17 bitri 
 |-  ( F : A -1-1-onto-> B <-> ( F Fn A /\ Fun `' F /\ ran F = B ) )