MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imass1 Unicode version

Theorem imass1 5172
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 5105 . . 3  |-  ( A 
C_  B  ->  ( A  |`  C )  C_  ( B  |`  C ) )
2 rnss 5031 . . 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 4824 . 2  |-  ( A
" C )  =  ran  ( A  |`  C )
5 df-ima 4824 . 2  |-  ( B
" C )  =  ran  ( B  |`  C )
63, 4, 53sstr4g 3325 1  |-  ( A 
C_  B  ->  ( A " C )  C_  ( B " C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3256   ran crn 4812    |` cres 4813   "cima 4814
This theorem is referenced by:  vdwnnlem1  13283  gsumzres  15437  gsumzadd  15447  gsum2d  15466  dprdfadd  15498  dprdres  15506  imasnopn  17636  imasncld  17637  imasncls  17638  tsmsres  18087  utoptop  18178  restutop  18181  ustuqtop3  18187  utopreg  18196  metustbl  18479  tdeglem4  19843  imadifxp  23874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824
  Copyright terms: Public domain W3C validator