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Theorem imass1 5359
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 5287 . . 3  |-  ( A 
C_  B  ->  ( A  |`  C )  C_  ( B  |`  C ) )
2 rnss 5220 . . 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 5001 . 2  |-  ( A
" C )  =  ran  ( A  |`  C )
5 df-ima 5001 . 2  |-  ( B
" C )  =  ran  ( B  |`  C )
63, 4, 53sstr4g 3530 1  |-  ( A 
C_  B  ->  ( A " C )  C_  ( B " C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    C_ wss 3461   ran crn 4989    |` cres 4990   "cima 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001
This theorem is referenced by:  vdwnnlem1  14597  gsumzresOLD  17117  gsumzaddOLD  17136  gsum2dOLD  17196  dprdfaddOLD  17262  dprdres  17270  imasnopn  20357  imasncld  20358  imasncls  20359  tsmsresOLD  20811  utoptop  20903  restutop  20906  ustuqtop3  20912  utopreg  20921  metustblOLD  21249  metustbl  21250  imadifxp  27672  esum2d  28322  eulerpartlemmf  28578  hess  38253
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