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Theorem dff1o2 5493
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 5278 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
2 df-f1 5276 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
3 df-fo 5277 . . . 4  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
42, 3anbi12i 678 . . 3  |-  ( ( F : A -1-1-> B  /\  F : A -onto-> B
)  <->  ( ( F : A --> B  /\  Fun  `' F )  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
5 anass 630 . . . 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 947 . . . . . 6  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
76anbi1i 676 . . . . 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 3243 . . . . . . . 8  |-  ( ran 
F  =  B  ->  ran  F  C_  B )
9 df-f 5275 . . . . . . . . 9  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
109biimpri 197 . . . . . . . 8  |-  ( ( F  Fn  A  /\  ran  F  C_  B )  ->  F : A --> B )
118, 10sylan2 460 . . . . . . 7  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  F : A --> B )
12113adant2 974 . . . . . 6  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  ->  F : A --> B )
1312pm4.71i 613 . . . . 5  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  /\  F : A --> B ) )
14 ancom 437 . . . . 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 269 . . . 4  |-  ( ( F : A --> B  /\  ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
165, 15bitri 240 . . 3  |-  ( ( ( F : A --> B  /\  Fun  `' F
)  /\  ( F  Fn  A  /\  ran  F  =  B ) )  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
174, 16bitri 240 . 2  |-  ( ( F : A -1-1-> B  /\  F : A -onto-> B
)  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
181, 17bitri 240 1  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    C_ wss 3165   `'ccnv 4704   ran crn 4706   Fun wfun 5265    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270
This theorem is referenced by:  dff1o3  5494  dff1o4  5496  f1orn  5498  tz7.49c  6474  fiint  7149  dfrelog  19939  adj1o  22490  esumc  23445  stoweidlem39  27891
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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