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

Theorem f1ores 5503
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 5460 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-> B )
2 f1f1orn 5499 . . 3  |-  ( ( F  |`  C ) : C -1-1-> B  ->  ( F  |`  C ) : C -1-1-onto-> ran  ( F  |`  C ) )
31, 2syl 15 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ran  ( F  |`  C ) )
4 df-ima 4718 . . 3  |-  ( F
" C )  =  ran  ( F  |`  C )
5 f1oeq3 5481 . . 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 203 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 176    /\ wa 358    = wceq 1632    C_ wss 3165   ran crn 4706    |` cres 4707   "cima 4708   -1-1->wf1 5268   -1-1-onto->wf1o 5270
This theorem is referenced by:  f1imacnv  5505  isores3  5848  isoini2  5852  f1imaeng  6937  f1imaen2g  6938  domunsncan  6978  php3  7063  ssfi  7099  infdifsn  7373  infxpenlem  7657  ackbij2lem2  7882  fin1a2lem6  8047  grothomex  8467  fsumss  12214  ackbijnn  12302  unbenlem  12971  eqgen  14686  gsumval3  15207  gsumzaddlem  15219  coe1mul2lem2  16361  tsmsf1o  17843  ovoliunlem1  18877  dvcnvrelem2  19381  logf1o2  20013  dvlog  20014  adjbd1o  22681  rinvf1o  23054  ballotlemfrc  23101  indf1ofs  23624  erdsze2lem2  23750  eupares  23914  ismtyres  26635  pwfi2f1o  27363  lindsmm  27401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
  Copyright terms: Public domain W3C validator