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Theorem resima 5003
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 5002 . . 3  |-  ( ( A  |`  B )  |`  B )  =  ( A  |`  B )
21rneqi 4921 . 2  |-  ran  (
( A  |`  B )  |`  B )  =  ran  ( A  |`  B )
3 df-ima 4718 . 2  |-  ( ( A  |`  B ) " B )  =  ran  ( ( A  |`  B )  |`  B )
4 df-ima 4718 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2326 1  |-  ( ( A  |`  B ) " B )  =  ( A " B )
Colors of variables: wff set class
Syntax hints:    = wceq 1632   ran crn 4706    |` cres 4707   "cima 4708
This theorem is referenced by:  isarep2  5348  f1imacnv  5505  foimacnv  5506  dffv2  5608  qtopres  17405  islindf4  27411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
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