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Theorem imass1 4750
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 4685 . . 3  |-  ( A 
C_  B  ->  ( A  |`  C )  C_  ( B  |`  C ) )
2 rnss 4612 . . 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 4404 . 2  |-  ( A
" C )  =  ran  ( A  |`  C )
5 df-ima 4404 . 2  |-  ( B
" C )  =  ran  ( B  |`  C )
63, 4, 53sstr4g 3049 1  |-  ( A 
C_  B  ->  ( A " C )  C_  ( B " C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 2982   ran crn 4392    |` cres 4393   "cima 4394
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404
This theorem is referenced by: (None)
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