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Theorem f1ores 5495
Description: The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)
Assertion
Ref Expression
f1ores  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )

Proof of Theorem f1ores
StepHypRef Expression
1 f1ssres 5449 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-> B )
2 f1f1orn 5491 . . 3  |-  ( ( F  |`  C ) : C -1-1-> B  ->  ( F  |`  C ) : C -1-1-onto-> ran  ( F  |`  C ) )
31, 2syl 14 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ran  ( F  |`  C ) )
4 df-ima 4657 . . 3  |-  ( F
" C )  =  ran  ( F  |`  C )
5 f1oeq3 5470 . . 3  |-  ( ( F " C )  =  ran  ( F  |`  C )  ->  (
( F  |`  C ) : C -1-1-onto-> ( F " C
)  <->  ( F  |`  C ) : C -1-1-onto-> ran  ( F  |`  C ) ) )
64, 5ax-mp 5 . 2  |-  ( ( F  |`  C ) : C -1-1-onto-> ( F " C
)  <->  ( F  |`  C ) : C -1-1-onto-> ran  ( F  |`  C ) )
73, 6sylibr 134 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    C_ wss 3144   ran crn 4645    |` cres 4646   "cima 4647   -1-1->wf1 5232   -1-1-onto->wf1o 5234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242
This theorem is referenced by:  f1imacnv  5497  f1oresrab  5702  isores3  5837  isoini2  5841  f1imaeng  6818  f1imaen2g  6819  preimaf1ofi  6980  endjusym  7125  dju1p1e2  7226  fisumss  11432  fprodssdc  11630  ssnnctlemct  12497  eqgen  13166
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