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Theorem foima 5345
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 4886 . 2  |-  ( F
" dom  F )  =  ran  F
2 fof 5340 . . . 4  |-  ( F : A -onto-> B  ->  F : A --> B )
3 fdm 5273 . . . 4  |-  ( F : A --> B  ->  dom  F  =  A )
42, 3syl 14 . . 3  |-  ( F : A -onto-> B  ->  dom  F  =  A )
54imaeq2d 4876 . 2  |-  ( F : A -onto-> B  -> 
( F " dom  F )  =  ( F
" A ) )
6 forn 5343 . 2  |-  ( F : A -onto-> B  ->  ran  F  =  B )
71, 5, 63eqtr3a 2194 1  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   dom cdm 4534   ran crn 4535   "cima 4537   -->wf 5114   -onto->wfo 5116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-fn 5121  df-f 5122  df-fo 5124
This theorem is referenced by:  foimacnv  5378  foima2  5646  fiintim  6810  fidcenumlemr  6836
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