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Theorem imadmres 5430
Description: The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
Assertion
Ref Expression
imadmres  |-  ( A
" dom  ( A  |`  B ) )  =  ( A " B
)

Proof of Theorem imadmres
StepHypRef Expression
1 resdmres 5429 . . 3  |-  ( A  |`  dom  ( A  |`  B ) )  =  ( A  |`  B )
21rneqi 5166 . 2  |-  ran  ( A  |`  dom  ( A  |`  B ) )  =  ran  ( A  |`  B )
3 df-ima 4953 . 2  |-  ( A
" dom  ( A  |`  B ) )  =  ran  ( A  |`  dom  ( A  |`  B ) )
4 df-ima 4953 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2490 1  |-  ( A
" dom  ( A  |`  B ) )  =  ( A " B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   dom cdm 4940   ran crn 4941    |` cres 4942   "cima 4943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-cnv 4948  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953
This theorem is referenced by:  ssimaex  5857  fnwelem  6789  imafi  7707  r0weon  8282  limsupgle  13059  kqdisj  19423
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