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Theorem resiima 5220
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 4891 . . 3  |-  ( (  _I  |`  A ) " B )  =  ran  ( (  _I  |`  A )  |`  B )
21a1i 11 . 2  |-  ( B 
C_  A  ->  (
(  _I  |`  A )
" B )  =  ran  ( (  _I  |`  A )  |`  B ) )
3 resabs1 5175 . . 3  |-  ( B 
C_  A  ->  (
(  _I  |`  A )  |`  B )  =  (  _I  |`  B )
)
43rneqd 5097 . 2  |-  ( B 
C_  A  ->  ran  ( (  _I  |`  A )  |`  B )  =  ran  (  _I  |`  B ) )
5 rnresi 5219 . . 3  |-  ran  (  _I  |`  B )  =  B
65a1i 11 . 2  |-  ( B 
C_  A  ->  ran  (  _I  |`  B )  =  B )
72, 4, 63eqtrd 2472 1  |-  ( B 
C_  A  ->  (
(  _I  |`  A )
" B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    C_ wss 3320    _I cid 4493   ran crn 4879    |` cres 4880   "cima 4881
This theorem is referenced by:  fipreima  7412  ssidcn  17319  idqtop  17738  fmid  17992  rrhre  24387  sitmcl  24663  islinds2  27260  lindsind2  27266  psgnunilem1  27393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891
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