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Theorem resima 5304
Description: A restriction to an image. (Contributed by NM, 29-Sep-2004.)
Assertion
Ref Expression
resima  |-  ( ( A  |`  B ) " B )  =  ( A " B )

Proof of Theorem resima
StepHypRef Expression
1 residm 5303 . . 3  |-  ( ( A  |`  B )  |`  B )  =  ( A  |`  B )
21rneqi 5227 . 2  |-  ran  (
( A  |`  B )  |`  B )  =  ran  ( A  |`  B )
3 df-ima 5012 . 2  |-  ( ( A  |`  B ) " B )  =  ran  ( ( A  |`  B )  |`  B )
4 df-ima 5012 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2506 1  |-  ( ( A  |`  B ) " B )  =  ( A " B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   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-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-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:  isarep2  5666  f1imacnv  5830  foimacnv  5831  dffv2  5938  islindf4  18640  qtopres  19934
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