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

Theorem fodmrnu 5475
Description: An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
Assertion
Ref Expression
fodmrnu  |-  ( ( F : A -onto-> B  /\  F : C -onto-> D
)  ->  ( A  =  C  /\  B  =  D ) )

Proof of Theorem fodmrnu
StepHypRef Expression
1 fofn 5469 . . 3  |-  ( F : A -onto-> B  ->  F  Fn  A )
2 fofn 5469 . . 3  |-  ( F : C -onto-> D  ->  F  Fn  C )
3 fndmu 5361 . . 3  |-  ( ( F  Fn  A  /\  F  Fn  C )  ->  A  =  C )
41, 2, 3syl2an 463 . 2  |-  ( ( F : A -onto-> B  /\  F : C -onto-> D
)  ->  A  =  C )
5 forn 5470 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
6 forn 5470 . . 3  |-  ( F : C -onto-> D  ->  ran  F  =  D )
75, 6sylan9req 2349 . 2  |-  ( ( F : A -onto-> B  /\  F : C -onto-> D
)  ->  B  =  D )
84, 7jca 518 1  |-  ( ( F : A -onto-> B  /\  F : C -onto-> D
)  ->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632   ran crn 4706    Fn wfn 5266   -onto->wfo 5269
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-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-fn 5274  df-f 5275  df-fo 5277
  Copyright terms: Public domain W3C validator