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Theorem fodmrnu 5459
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 5453 . . 3  |-  ( F : A -onto-> B  ->  F  Fn  A )
2 fofn 5453 . . 3  |-  ( F : C -onto-> D  ->  F  Fn  C )
3 fndmu 5345 . . 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 5454 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
6 forn 5454 . . 3  |-  ( F : C -onto-> D  ->  ran  F  =  D )
75, 6sylan9req 2336 . 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 1623   ran crn 4690    Fn wfn 5250   -onto->wfo 5253
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-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-fn 5258  df-f 5259  df-fo 5261
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