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Theorem foima 5472
Description: The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
Assertion
Ref Expression
foima  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )

Proof of Theorem foima
StepHypRef Expression
1 imadmrn 5040 . 2  |-  ( F
" dom  F )  =  ran  F
2 fof 5467 . . . 4  |-  ( F : A -onto-> B  ->  F : A --> B )
3 fdm 5409 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
42, 3syl 15 . . 3  |-  ( F : A -onto-> B  ->  dom  F  =  A )
54imaeq2d 5028 . 2  |-  ( F : A -onto-> B  -> 
( F " dom  F )  =  ( F
" A ) )
6 forn 5470 . 2  |-  ( F : A -onto-> B  ->  ran  F  =  B )
71, 5, 63eqtr3a 2352 1  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   dom cdm 4705   ran crn 4706   "cima 4708   -->wf 5267   -onto->wfo 5269
This theorem is referenced by:  foimacnv  5506  domunfican  7145  fiint  7149  fodomfi  7151  cantnflt2  7390  cantnfp1lem3  7398  enfin1ai  8026  dprdf1o  15283  cncmp  17135  cmpfi  17151  cnconn  17164  qtopval2  17403  elfm3  17661  rnelfm  17664  fmfnfmlem2  17666  fmfnfm  17669  pjordi  22769  eupath2  23919  ovoliunnfl  25001  ismtybndlem  26633  kelac1  27264  gicabl  27366  lmimlbs  27409
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-eu 2160  df-mo 2161  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-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fn 5274  df-f 5275  df-fo 5277
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