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Theorem imadmres 5122
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 5121 . . 3  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  |`  B )
21rneqi 4856 . 2  |-  ran  ( A  |`  dom  ( A  |`  B ) )  =  ran  ( A  |`  B )
3 df-ima 4640 . 2  |-  ( A
" dom  ( A  |`  B ) )  =  ran  ( A  |`  dom  ( A  |`  B ) )
4 df-ima 4640 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2208 1  |-  ( A
" dom  ( A  |`  B ) )  =  ( A " B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1353   dom cdm 4627   ran crn 4628    |` cres 4629   "cima 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640
This theorem is referenced by:  ssimaex  5578
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