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Theorem imass1 4807
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 4739 . . 3  |-  ( A 
C_  B  ->  ( A  |`  C )  C_  ( B  |`  C ) )
2 rnss 4665 . . 3  |-  ( ( A  |`  C )  C_  ( B  |`  C )  ->  ran  ( A  |`  C )  C_  ran  ( B  |`  C ) )
31, 2syl 14 . 2  |-  ( A 
C_  B  ->  ran  ( A  |`  C ) 
C_  ran  ( B  |`  C ) )
4 df-ima 4451 . 2  |-  ( A
" C )  =  ran  ( A  |`  C )
5 df-ima 4451 . 2  |-  ( B
" C )  =  ran  ( B  |`  C )
63, 4, 53sstr4g 3067 1  |-  ( A 
C_  B  ->  ( A " C )  C_  ( B " C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 2999   ran crn 4439    |` cres 4440   "cima 4441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451
This theorem is referenced by: (None)
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