MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1ores Unicode version

Theorem f1ores 5648
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 5605 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-> B )
2 f1f1orn 5644 . . 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 4850 . . 3  |-  ( F
" C )  =  ran  ( F  |`  C )
5 f1oeq3 5626 . . 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 3280   ran crn 4838    |` cres 4839   "cima 4840   -1-1->wf1 5410   -1-1-onto->wf1o 5412
This theorem is referenced by:  f1imacnv  5650  isores3  6014  isoini2  6018  f1imaeng  7126  f1imaen2g  7127  domunsncan  7167  php3  7252  ssfi  7288  infdifsn  7567  infxpenlem  7851  ackbij2lem2  8076  fin1a2lem6  8241  grothomex  8660  fsumss  12474  ackbijnn  12562  unbenlem  13231  eqgen  14948  gsumval3  15469  gsumzaddlem  15481  coe1mul2lem2  16616  tsmsf1o  18127  ovoliunlem1  19351  dvcnvrelem2  19855  logf1o2  20494  dvlog  20495  eupares  21650  adjbd1o  23541  rinvf1o  23995  indf1ofs  24376  ballotlemfrc  24737  erdsze2lem2  24843  fprodss  25227  ismtyres  26407  pwfi2f1o  27128  lindsmm  27166
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420
  Copyright terms: Public domain W3C validator