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Theorem dfrn4 5046
Description: Range defined in terms of image. (Contributed by NM, 14-May-2008.)
Assertion
Ref Expression
dfrn4  |-  ran  A  =  ( A " _V )

Proof of Theorem dfrn4
StepHypRef Expression
1 df-ima 4599 . 2  |-  ( A
" _V )  =  ran  ( A  |`  _V )
2 rnresv 5045 . 2  |-  ran  ( A  |`  _V )  =  ran  A
31, 2eqtr2i 2179 1  |-  ran  A  =  ( A " _V )
Colors of variables: wff set class
Syntax hints:    = wceq 1335   _Vcvv 2712   ran crn 4587    |` cres 4588   "cima 4589
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593  df-cnv 4594  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599
This theorem is referenced by:  dmmpt  5081  ctssdccl  7055
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