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Theorem imass2 5362
Description: Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
imass2  |-  ( A 
C_  B  ->  ( C " A )  C_  ( C " B ) )

Proof of Theorem imass2
StepHypRef Expression
1 ssres2 5290 . . 3  |-  ( A 
C_  B  ->  ( C  |`  A )  C_  ( C  |`  B ) )
2 rnss 5221 . . 3  |-  ( ( C  |`  A )  C_  ( C  |`  B )  ->  ran  ( C  |`  A )  C_  ran  ( C  |`  B ) )
31, 2syl 16 . 2  |-  ( A 
C_  B  ->  ran  ( C  |`  A ) 
C_  ran  ( C  |`  B ) )
4 df-ima 5002 . 2  |-  ( C
" A )  =  ran  ( C  |`  A )
5 df-ima 5002 . 2  |-  ( C
" B )  =  ran  ( C  |`  B )
63, 4, 53sstr4g 3530 1  |-  ( A 
C_  B  ->  ( C " A )  C_  ( C " B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    C_ wss 3461   ran crn 4990    |` cres 4991   "cima 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002
This theorem is referenced by:  funimass1  5651  funimass2  5652  fvimacnv  5987  f1imass  6157  ecinxp  7388  sbthlem1  7629  sbthlem2  7630  php3  7705  ordtypelem2  7947  mapfienOLD  8141  tcrank  8305  limsupgord  13277  isercoll  13472  isacs1i  15036  gsumzf1o  16896  gsumzf1oOLD  16899  dprdres  17054  dprd2da  17070  dmdprdsplit2lem  17073  lmhmlsp  17674  f1lindf  18835  iscnp4  19742  cnpco  19746  cncls2i  19749  cnntri  19750  cnrest2  19765  cnpresti  19767  cnprest  19768  1stcfb  19924  xkococnlem  20138  qtopval2  20175  tgqtop  20191  qtoprest  20196  kqdisj  20211  regr1lem  20218  kqreglem1  20220  kqreglem2  20221  kqnrmlem1  20222  kqnrmlem2  20223  nrmhmph  20273  fbasrn  20363  elfm2  20427  fmfnfmlem1  20433  fmco  20440  flffbas  20474  cnpflf2  20479  cnextcn  20545  metcnp3  21021  metusttoOLD  21038  metustto  21039  cfilucfilOLD  21050  cfilucfil  21051  uniioombllem3  21972  dyadmbllem  21986  mbfconstlem  22014  i1fima2  22064  itg2gt0  22145  ellimc3  22261  limcflf  22263  limcresi  22267  limciun  22276  lhop  22395  ig1peu  22550  ig1pdvds  22555  psercnlem2  22797  dvloglem  23007  efopn  23017  txomap  27815  tpr2rico  27872  cvmsss2  28697  cvmopnlem  28701  cvmliftmolem1  28704  cvmliftlem15  28721  cvmlift2lem9  28734  nofulllem3  29440  dvtan  30041  heibor1lem  30281  isnumbasabl  31031  isnumbasgrp  31032  dfacbasgrp  31033  limccog  31580
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