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Theorem resima 5125
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 5124 . . 3  |-  ( ( A  |`  B )  |`  B )  =  ( A  |`  B )
21rneqi 5049 . 2  |-  ran  (
( A  |`  B )  |`  B )  =  ran  ( A  |`  B )
3 df-ima 4835 . 2  |-  ( ( A  |`  B ) " B )  =  ran  ( ( A  |`  B )  |`  B )
4 df-ima 4835 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2441 1  |-  ( ( A  |`  B ) " B )  =  ( A " B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   ran crn 4823    |` cres 4824   "cima 4825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-rel 4829  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835
This theorem is referenced by:  isarep2  5648  f1imacnv  5814  foimacnv  5815  dffv2  5921  islindf4  19163  qtopres  20489
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