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Theorem rnss 4816
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 4759 . . 3  |-  ( A 
C_  B  ->  `' A  C_  `' B )
2 dmss 4785 . . 3  |-  ( `' A  C_  `' B  ->  dom  `' A  C_  dom  `' B )
31, 2syl 14 . 2  |-  ( A 
C_  B  ->  dom  `' A  C_  dom  `' B
)
4 df-rn 4597 . 2  |-  ran  A  =  dom  `' A
5 df-rn 4597 . 2  |-  ran  B  =  dom  `' B
63, 4, 53sstr4g 3171 1  |-  ( A 
C_  B  ->  ran  A 
C_  ran  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3102   `'ccnv 4585   dom cdm 4586   ran crn 4587
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4594  df-dm 4596  df-rn 4597
This theorem is referenced by:  imass1  4961  imass2  4962  rnxpss2  5019  ssxpbm  5021  ssxp2  5023  ssrnres  5028  funssxp  5339  fssres  5345  dff2  5611  fliftf  5749  1stcof  6111  2ndcof  6112  smores  6239  tfrcllembfn  6304  caserel  7031  frecuzrdgtcl  10311  lmss  12657  txss12  12677  txbasval  12678
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