ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  imass2 Unicode version

Theorem imass2 4883
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 4814 . . 3  |-  ( A 
C_  B  ->  ( C  |`  A )  C_  ( C  |`  B ) )
2 rnss 4737 . . 3  |-  ( ( C  |`  A )  C_  ( C  |`  B )  ->  ran  ( C  |`  A )  C_  ran  ( C  |`  B ) )
31, 2syl 14 . 2  |-  ( A 
C_  B  ->  ran  ( C  |`  A ) 
C_  ran  ( C  |`  B ) )
4 df-ima 4520 . 2  |-  ( C
" A )  =  ran  ( C  |`  A )
5 df-ima 4520 . 2  |-  ( C
" B )  =  ran  ( C  |`  B )
63, 4, 53sstr4g 3108 1  |-  ( A 
C_  B  ->  ( C " A )  C_  ( C " B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3039   ran crn 4508    |` cres 4509   "cima 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520
This theorem is referenced by:  funimass1  5168  funimass2  5169  fvimacnv  5501  f1imass  5641  ecinxp  6470  sbthlem1  6811  sbthlem2  6812  iscnp4  12293  cnptopco  12297  cnntri  12299  cnrest2  12311  cnptopresti  12313  cnptoprest  12314  metcnp3  12586
  Copyright terms: Public domain W3C validator