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Theorem f1ores 5630
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 5587 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-> B )
2 f1f1orn 5626 . . 3  |-  ( ( F  |`  C ) : C -1-1-> B  ->  ( F  |`  C ) : C -1-1-onto-> ran  ( F  |`  C ) )
31, 2syl 16 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ran  ( F  |`  C ) )
4 df-ima 4832 . . 3  |-  ( F
" C )  =  ran  ( F  |`  C )
5 f1oeq3 5608 . . 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 8 . 2  |-  ( ( F  |`  C ) : C -1-1-onto-> ( F " C
)  <->  ( F  |`  C ) : C -1-1-onto-> ran  ( F  |`  C ) )
73, 6sylibr 204 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    <-> wb 177    /\ wa 359    = wceq 1649    C_ wss 3264   ran crn 4820    |` cres 4821   "cima 4822   -1-1->wf1 5392   -1-1-onto->wf1o 5394
This theorem is referenced by:  f1imacnv  5632  isores3  5995  isoini2  5999  f1imaeng  7104  f1imaen2g  7105  domunsncan  7145  php3  7230  ssfi  7266  infdifsn  7545  infxpenlem  7829  ackbij2lem2  8054  fin1a2lem6  8219  grothomex  8638  fsumss  12447  ackbijnn  12535  unbenlem  13204  eqgen  14921  gsumval3  15442  gsumzaddlem  15454  coe1mul2lem2  16589  tsmsf1o  18096  ovoliunlem1  19266  dvcnvrelem2  19770  logf1o2  20409  dvlog  20410  eupares  21546  adjbd1o  23437  rinvf1o  23886  indf1ofs  24220  ballotlemfrc  24564  erdsze2lem2  24670  fprodss  25054  ismtyres  26209  pwfi2f1o  26930  lindsmm  26968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402
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