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Theorem imass1 5232
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
Assertion
Ref Expression
imass1  |-  ( A 
C_  B  ->  ( A " C )  C_  ( B " C ) )

Proof of Theorem imass1
StepHypRef Expression
1 ssres 5165 . . 3  |-  ( A 
C_  B  ->  ( A  |`  C )  C_  ( B  |`  C ) )
2 rnss 5091 . . 3  |-  ( ( A  |`  C )  C_  ( B  |`  C )  ->  ran  ( A  |`  C )  C_  ran  ( B  |`  C ) )
31, 2syl 16 . 2  |-  ( A 
C_  B  ->  ran  ( A  |`  C ) 
C_  ran  ( B  |`  C ) )
4 df-ima 4884 . 2  |-  ( A
" C )  =  ran  ( A  |`  C )
5 df-ima 4884 . 2  |-  ( B
" C )  =  ran  ( B  |`  C )
63, 4, 53sstr4g 3382 1  |-  ( A 
C_  B  ->  ( A " C )  C_  ( B " C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3313   ran crn 4872    |` cres 4873   "cima 4874
This theorem is referenced by:  vdwnnlem1  13356  gsumzres  15510  gsumzadd  15520  gsum2d  15539  dprdfadd  15571  dprdres  15579  imasnopn  17715  imasncld  17716  imasncls  17717  tsmsres  18166  utoptop  18257  restutop  18260  ustuqtop3  18266  utopreg  18275  metustblOLD  18603  metustbl  18604  tdeglem4  19976  imadifxp  24031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-cnv 4879  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884
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