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Theorem fnima 5132
Description: The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fnima  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )

Proof of Theorem fnima
StepHypRef Expression
1 df-ima 4451 . 2  |-  ( F
" A )  =  ran  ( F  |`  A )
2 fnresdm 5123 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
32rneqd 4664 . 2  |-  ( F  Fn  A  ->  ran  ( F  |`  A )  =  ran  F )
41, 3syl5eq 2132 1  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   ran crn 4439    |` cres 4440   "cima 4441    Fn wfn 5010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-fun 5017  df-fn 5018
This theorem is referenced by:  f1finf1o  6656
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