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Theorem f1ores 5141
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 5099 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-> B )
2 f1f1orn 5137 . . 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 4358 . . 3  |-  ( F
" C )  =  ran  ( F  |`  C )
5 f1oeq3 5119 . . 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 7 . 2  |-  ( ( F  |`  C ) : C -1-1-onto-> ( F " C
)  <->  ( F  |`  C ) : C -1-1-onto-> ran  ( F  |`  C ) )
73, 6sylibr 137 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 97    <-> wb 98    = wceq 1243    C_ wss 2917   ran crn 4346    |` cres 4347   "cima 4348   -1-1->wf1 4899   -1-1-onto->wf1o 4901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909
This theorem is referenced by:  f1imacnv  5143  f1oresrab  5329  isores3  5455  isoini2  5458  f1imaeng  6272  f1imaen2g  6273
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