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Theorem dff1o2 5306
Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
dff1o2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )

Proof of Theorem dff1o2
StepHypRef Expression
1 df-f1o 5066 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
2 df-f1 5064 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
3 df-fo 5065 . . . 4  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
42, 3anbi12i 451 . . 3  |-  ( ( F : A -1-1-> B  /\  F : A -onto-> B
)  <->  ( ( F : A --> B  /\  Fun  `' F )  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
5 anass 396 . . . 4  |-  ( ( ( F : A --> B  /\  Fun  `' F
)  /\  ( F  Fn  A  /\  ran  F  =  B ) )  <->  ( F : A --> B  /\  ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) ) ) )
6 3anan12 942 . . . . . 6  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
76anbi1i 449 . . . . 5  |-  ( ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  /\  F : A --> B )  <->  ( ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) )  /\  F : A
--> B ) )
8 eqimss 3101 . . . . . . . 8  |-  ( ran 
F  =  B  ->  ran  F  C_  B )
9 df-f 5063 . . . . . . . . 9  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
109biimpri 132 . . . . . . . 8  |-  ( ( F  Fn  A  /\  ran  F  C_  B )  ->  F : A --> B )
118, 10sylan2 282 . . . . . . 7  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  F : A --> B )
12113adant2 968 . . . . . 6  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  ->  F : A --> B )
1312pm4.71i 386 . . . . 5  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  /\  F : A --> B ) )
14 ancom 264 . . . . 5  |-  ( ( F : A --> B  /\  ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )  <->  ( ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) )  /\  F : A
--> B ) )
157, 13, 143bitr4ri 212 . . . 4  |-  ( ( F : A --> B  /\  ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
165, 15bitri 183 . . 3  |-  ( ( ( F : A --> B  /\  Fun  `' F
)  /\  ( F  Fn  A  /\  ran  F  =  B ) )  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
174, 16bitri 183 . 2  |-  ( ( F : A -1-1-> B  /\  F : A -onto-> B
)  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
181, 17bitri 183 1  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    /\ w3a 930    = wceq 1299    C_ wss 3021   `'ccnv 4476   ran crn 4478   Fun wfun 5053    Fn wfn 5054   -->wf 5055   -1-1->wf1 5056   -onto->wfo 5057   -1-1-onto->wf1o 5058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-in 3027  df-ss 3034  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066
This theorem is referenced by:  dff1o3  5307  dff1o4  5309  f1orn  5311  dif1en  6702  fiintim  6746
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