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Theorem imadmres 5165
Description: The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
Assertion
Ref Expression
imadmres  |-  ( A
" dom  ( A  |`  B ) )  =  ( A " B
)

Proof of Theorem imadmres
StepHypRef Expression
1 resdmres 5164 . . 3  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  |`  B )
21rneqi 4905 . 2  |-  ran  ( A  |`  dom  ( A  |`  B ) )  =  ran  ( A  |`  B )
3 df-ima 4702 . 2  |-  ( A
" dom  ( A  |`  B ) )  =  ran  ( A  |`  dom  ( A  |`  B ) )
4 df-ima 4702 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2313 1  |-  ( A
" dom  ( A  |`  B ) )  =  ( A " B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692
This theorem is referenced by:  ssimaex  5584  fnwelem  6230  imafi  7148  r0weon  7640  limsupgle  11951  kqdisj  17423
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-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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