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Theorem dff1o2 5612
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 5394 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
2 df-f1 5392 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
3 df-fo 5393 . . . 4  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
42, 3anbi12i 679 . . 3  |-  ( ( F : A -1-1-> B  /\  F : A -onto-> B
)  <->  ( ( F : A --> B  /\  Fun  `' F )  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
5 anass 631 . . . 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 949 . . . . . 6  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )
76anbi1i 677 . . . . 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 3336 . . . . . . . 8  |-  ( ran 
F  =  B  ->  ran  F  C_  B )
9 df-f 5391 . . . . . . . . 9  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
109biimpri 198 . . . . . . . 8  |-  ( ( F  Fn  A  /\  ran  F  C_  B )  ->  F : A --> B )
118, 10sylan2 461 . . . . . . 7  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  F : A --> B )
12113adant2 976 . . . . . 6  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  ->  F : A --> B )
1312pm4.71i 614 . . . . 5  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  /\  F : A --> B ) )
14 ancom 438 . . . . 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 270 . . . 4  |-  ( ( F : A --> B  /\  ( Fun  `' F  /\  ( F  Fn  A  /\  ran  F  =  B ) ) )  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
165, 15bitri 241 . . 3  |-  ( ( ( F : A --> B  /\  Fun  `' F
)  /\  ( F  Fn  A  /\  ran  F  =  B ) )  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
174, 16bitri 241 . 2  |-  ( ( F : A -1-1-> B  /\  F : A -onto-> B
)  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
181, 17bitri 241 1  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    C_ wss 3256   `'ccnv 4810   ran crn 4812   Fun wfun 5381    Fn wfn 5382   -->wf 5383   -1-1->wf1 5384   -onto->wfo 5385   -1-1-onto->wf1o 5386
This theorem is referenced by:  dff1o3  5613  dff1o4  5615  f1orn  5617  tz7.49c  6632  fiint  7312  dfrelog  20323  adj1o  23238  esumc  24235  stoweidlem39  27449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-in 3263  df-ss 3270  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394
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