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Theorem resiima 4683
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 4358 . . 3  |-  ( (  _I  |`  A ) " B )  =  ran  ( (  _I  |`  A )  |`  B )
21a1i 9 . 2  |-  ( B 
C_  A  ->  (
(  _I  |`  A )
" B )  =  ran  ( (  _I  |`  A )  |`  B ) )
3 resabs1 4640 . . 3  |-  ( B 
C_  A  ->  (
(  _I  |`  A )  |`  B )  =  (  _I  |`  B )
)
43rneqd 4563 . 2  |-  ( B 
C_  A  ->  ran  ( (  _I  |`  A )  |`  B )  =  ran  (  _I  |`  B ) )
5 rnresi 4682 . . 3  |-  ran  (  _I  |`  B )  =  B
65a1i 9 . 2  |-  ( B 
C_  A  ->  ran  (  _I  |`  B )  =  B )
72, 4, 63eqtrd 2076 1  |-  ( B 
C_  A  ->  (
(  _I  |`  A )
" B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    C_ wss 2917    _I cid 4025   ran crn 4346    |` cres 4347   "cima 4348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358
This theorem is referenced by: (None)
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