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Theorem rnss 4737
Description: Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
rnss  |-  ( A 
C_  B  ->  ran  A 
C_  ran  B )

Proof of Theorem rnss
StepHypRef Expression
1 cnvss 4680 . . 3  |-  ( A 
C_  B  ->  `' A  C_  `' B )
2 dmss 4706 . . 3  |-  ( `' A  C_  `' B  ->  dom  `' A  C_  dom  `' B )
31, 2syl 14 . 2  |-  ( A 
C_  B  ->  dom  `' A  C_  dom  `' B
)
4 df-rn 4518 . 2  |-  ran  A  =  dom  `' A
5 df-rn 4518 . 2  |-  ran  B  =  dom  `' B
63, 4, 53sstr4g 3108 1  |-  ( A 
C_  B  ->  ran  A 
C_  ran  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3039   `'ccnv 4506   dom cdm 4507   ran crn 4508
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-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  imass1  4882  imass2  4883  rnxpss2  4940  ssxpbm  4942  ssxp2  4944  ssrnres  4949  funssxp  5260  fssres  5266  dff2  5530  fliftf  5666  1stcof  6027  2ndcof  6028  smores  6155  tfrcllembfn  6220  caserel  6938  frecuzrdgtcl  10136  lmss  12321  txss12  12341  txbasval  12342
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