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Theorem foima 5617
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 5174 . 2  |-  ( F
" dom  F )  =  ran  F
2 fof 5612 . . . 4  |-  ( F : A -onto-> B  ->  F : A --> B )
3 fdm 5554 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
42, 3syl 16 . . 3  |-  ( F : A -onto-> B  ->  dom  F  =  A )
54imaeq2d 5162 . 2  |-  ( F : A -onto-> B  -> 
( F " dom  F )  =  ( F
" A ) )
6 forn 5615 . 2  |-  ( F : A -onto-> B  ->  ran  F  =  B )
71, 5, 63eqtr3a 2460 1  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   dom cdm 4837   ran crn 4838   "cima 4840   -->wf 5409   -onto->wfo 5411
This theorem is referenced by:  foimacnv  5651  domunfican  7338  fiint  7342  fodomfi  7344  cantnflt2  7584  cantnfp1lem3  7592  enfin1ai  8220  dprdf1o  15545  cncmp  17409  cmpfi  17425  cnconn  17438  qtopval2  17681  elfm3  17935  rnelfm  17938  fmfnfmlem2  17940  fmfnfm  17943  eupath2  21655  pjordi  23629  ovoliunnfl  26147  voliunnfl  26149  volsupnfl  26150  ismtybndlem  26405  kelac1  27029  gicabl  27131  lmimlbs  27174
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-eu 2258  df-mo 2259  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-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fn 5416  df-f 5417  df-fo 5419
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