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Theorem resiima 4904
Description: The image of a restriction of the identity function. (Contributed by FL, 31-Dec-2006.)
Assertion
Ref Expression
resiima  |-  ( B 
C_  A  ->  (
(  _I  |`  A )
" B )  =  B )

Proof of Theorem resiima
StepHypRef Expression
1 df-ima 4559 . . 3  |-  ( (  _I  |`  A ) " B )  =  ran  ( (  _I  |`  A )  |`  B )
21a1i 9 . 2  |-  ( B 
C_  A  ->  (
(  _I  |`  A )
" B )  =  ran  ( (  _I  |`  A )  |`  B ) )
3 resabs1 4855 . . 3  |-  ( B 
C_  A  ->  (
(  _I  |`  A )  |`  B )  =  (  _I  |`  B )
)
43rneqd 4775 . 2  |-  ( B 
C_  A  ->  ran  ( (  _I  |`  A )  |`  B )  =  ran  (  _I  |`  B ) )
5 rnresi 4903 . . 3  |-  ran  (  _I  |`  B )  =  B
65a1i 9 . 2  |-  ( B 
C_  A  ->  ran  (  _I  |`  B )  =  B )
72, 4, 63eqtrd 2177 1  |-  ( B 
C_  A  ->  (
(  _I  |`  A )
" B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332    C_ wss 3075    _I cid 4217   ran crn 4547    |` cres 4548   "cima 4549
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559
This theorem is referenced by:  ssidcn  12416
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