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Theorem fodmrnu 5323
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 5317 . . 3  |-  ( F : A -onto-> B  ->  F  Fn  A )
2 fofn 5317 . . 3  |-  ( F : C -onto-> D  ->  F  Fn  C )
3 fndmu 5194 . . 3  |-  ( ( F  Fn  A  /\  F  Fn  C )  ->  A  =  C )
41, 2, 3syl2an 287 . 2  |-  ( ( F : A -onto-> B  /\  F : C -onto-> D
)  ->  A  =  C )
5 forn 5318 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
6 forn 5318 . . 3  |-  ( F : C -onto-> D  ->  ran  F  =  D )
75, 6sylan9req 2171 . 2  |-  ( ( F : A -onto-> B  /\  F : C -onto-> D
)  ->  B  =  D )
84, 7jca 304 1  |-  ( ( F : A -onto-> B  /\  F : C -onto-> D
)  ->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316   ran crn 4510    Fn wfn 5088   -onto->wfo 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-in 3047  df-ss 3054  df-fn 5096  df-f 5097  df-fo 5099
This theorem is referenced by: (None)
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