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Theorem resiima 5341
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 5002 . . 3  |-  ( (  _I  |`  A ) " B )  =  ran  ( (  _I  |`  A )  |`  B )
21a1i 11 . 2  |-  ( B 
C_  A  ->  (
(  _I  |`  A )
" B )  =  ran  ( (  _I  |`  A )  |`  B ) )
3 resabs1 5292 . . 3  |-  ( B 
C_  A  ->  (
(  _I  |`  A )  |`  B )  =  (  _I  |`  B )
)
43rneqd 5220 . 2  |-  ( B 
C_  A  ->  ran  ( (  _I  |`  A )  |`  B )  =  ran  (  _I  |`  B ) )
5 rnresi 5340 . . 3  |-  ran  (  _I  |`  B )  =  B
65a1i 11 . 2  |-  ( B 
C_  A  ->  ran  (  _I  |`  B )  =  B )
72, 4, 63eqtrd 2488 1  |-  ( B 
C_  A  ->  (
(  _I  |`  A )
" B )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    C_ wss 3461    _I cid 4780   ran crn 4990    |` cres 4991   "cima 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002
This theorem is referenced by:  fipreima  7828  psgnunilem1  16497  islinds2  18826  lindsind2  18832  ssidcn  19734  idqtop  20185  fmid  20439  rrhre  27977
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