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Theorem imadmres 5490
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 5489 . . 3  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  |`  B )
21rneqi 5220 . 2  |-  ran  ( A  |`  dom  ( A  |`  B ) )  =  ran  ( A  |`  B )
3 df-ima 5005 . 2  |-  ( A
" dom  ( A  |`  B ) )  =  ran  ( A  |`  dom  ( A  |`  B ) )
4 df-ima 5005 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2499 1  |-  ( A
" dom  ( A  |`  B ) )  =  ( A " B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374   dom cdm 4992   ran crn 4993    |` cres 4994   "cima 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005
This theorem is referenced by:  ssimaex  5923  fnwelem  6888  imafi  7802  r0weon  8379  limsupgle  13249  kqdisj  19961
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