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Theorem resiima 5351
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 5012 . . 3  |-  ( (  _I  |`  A ) " B )  =  ran  ( (  _I  |`  A )  |`  B )
21a1i 11 . 2  |-  ( B 
C_  A  ->  (
(  _I  |`  A )
" B )  =  ran  ( (  _I  |`  A )  |`  B ) )
3 resabs1 5302 . . 3  |-  ( B 
C_  A  ->  (
(  _I  |`  A )  |`  B )  =  (  _I  |`  B )
)
43rneqd 5230 . 2  |-  ( B 
C_  A  ->  ran  ( (  _I  |`  A )  |`  B )  =  ran  (  _I  |`  B ) )
5 rnresi 5350 . . 3  |-  ran  (  _I  |`  B )  =  B
65a1i 11 . 2  |-  ( B 
C_  A  ->  ran  (  _I  |`  B )  =  B )
72, 4, 63eqtrd 2512 1  |-  ( B 
C_  A  ->  (
(  _I  |`  A )
" B )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    C_ wss 3476    _I cid 4790   ran crn 5000    |` cres 5001   "cima 5002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012
This theorem is referenced by:  fipreima  7826  psgnunilem1  16324  islinds2  18643  lindsind2  18649  ssidcn  19550  idqtop  19970  fmid  20224  rrhre  27663
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