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Theorem imadmres 5268
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 5267 . . 3  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  |`  B )
21rneqi 5008 . 2  |-  ran  ( A  |`  dom  ( A  |`  B ) )  =  ran  ( A  |`  B )
3 df-ima 4805 . 2  |-  ( A
" dom  ( A  |`  B ) )  =  ran  ( A  |`  dom  ( A  |`  B ) )
4 df-ima 4805 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2396 1  |-  ( A
" dom  ( A  |`  B ) )  =  ( A " B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1647   dom cdm 4792   ran crn 4793    |` cres 4794   "cima 4795
This theorem is referenced by:  ssimaex  5691  fnwelem  6358  imafi  7295  r0weon  7787  limsupgle  12158  kqdisj  17640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-xp 4798  df-rel 4799  df-cnv 4800  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805
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