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Theorem fodmrnu 5661
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 5655 . . 3  |-  ( F : A -onto-> B  ->  F  Fn  A )
2 fofn 5655 . . 3  |-  ( F : C -onto-> D  ->  F  Fn  C )
3 fndmu 5546 . . 3  |-  ( ( F  Fn  A  /\  F  Fn  C )  ->  A  =  C )
41, 2, 3syl2an 464 . 2  |-  ( ( F : A -onto-> B  /\  F : C -onto-> D
)  ->  A  =  C )
5 forn 5656 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
6 forn 5656 . . 3  |-  ( F : C -onto-> D  ->  ran  F  =  D )
75, 6sylan9req 2489 . 2  |-  ( ( F : A -onto-> B  /\  F : C -onto-> D
)  ->  B  =  D )
84, 7jca 519 1  |-  ( ( F : A -onto-> B  /\  F : C -onto-> D
)  ->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652   ran crn 4879    Fn wfn 5449   -onto->wfo 5452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-in 3327  df-ss 3334  df-fn 5457  df-f 5458  df-fo 5460
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