MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dff1o2 Unicode version

Theorem dff1o2 5477
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 5262 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
2 df-f1 5260 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
3 df-fo 5261 . . . 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 3230 . . . . . . . 8  |-  ( ran 
F  =  B  ->  ran  F  C_  B )
9 df-f 5259 . . . . . . . . 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 1623    C_ wss 3152   `'ccnv 4688   ran crn 4690   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254
This theorem is referenced by:  dff1o3  5478  dff1o4  5480  f1orn  5482  tz7.49c  6458  fiint  7133  dfrelog  19923  adj1o  22474  esumc  23430  stoweidlem39  27788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
  Copyright terms: Public domain W3C validator