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Theorem foima 5456
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 5024 . 2  |-  ( F
" dom  F )  =  ran  F
2 fof 5451 . . . 4  |-  ( F : A -onto-> B  ->  F : A --> B )
3 fdm 5393 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
42, 3syl 15 . . 3  |-  ( F : A -onto-> B  ->  dom  F  =  A )
54imaeq2d 5012 . 2  |-  ( F : A -onto-> B  -> 
( F " dom  F )  =  ( F
" A ) )
6 forn 5454 . 2  |-  ( F : A -onto-> B  ->  ran  F  =  B )
71, 5, 63eqtr3a 2339 1  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   dom cdm 4689   ran crn 4690   "cima 4692   -->wf 5251   -onto->wfo 5253
This theorem is referenced by:  foimacnv  5490  domunfican  7129  fiint  7133  fodomfi  7135  cantnflt2  7374  cantnfp1lem3  7382  enfin1ai  8010  dprdf1o  15267  cncmp  17119  cmpfi  17135  cnconn  17148  qtopval2  17387  elfm3  17645  rnelfm  17648  fmfnfmlem2  17650  fmfnfm  17653  pjordi  22753  eupath2  23904  ismtybndlem  26530  kelac1  27161  gicabl  27263  lmimlbs  27306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fn 5258  df-f 5259  df-fo 5261
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