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Theorem resima 5153
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 5152 . . 3  |-  ( ( A  |`  B )  |`  B )  =  ( A  |`  B )
21rneqi 5077 . 2  |-  ran  (
( A  |`  B )  |`  B )  =  ran  ( A  |`  B )
3 df-ima 4863 . 2  |-  ( ( A  |`  B ) " B )  =  ran  ( ( A  |`  B )  |`  B )
4 df-ima 4863 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2461 1  |-  ( ( A  |`  B ) " B )  =  ( A " B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   ran crn 4851    |` cres 4852   "cima 4853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-xp 4856  df-rel 4857  df-cnv 4858  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863
This theorem is referenced by:  isarep2  5678  f1imacnv  5844  foimacnv  5845  dffv2  5951  islindf4  19383  qtopres  20700
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