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Theorem rnss 4841
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 4784 . . 3  |-  ( A 
C_  B  ->  `' A  C_  `' B )
2 dmss 4810 . . 3  |-  ( `' A  C_  `' B  ->  dom  `' A  C_  dom  `' B )
31, 2syl 14 . 2  |-  ( A 
C_  B  ->  dom  `' A  C_  dom  `' B
)
4 df-rn 4622 . 2  |-  ran  A  =  dom  `' A
5 df-rn 4622 . 2  |-  ran  B  =  dom  `' B
63, 4, 53sstr4g 3190 1  |-  ( A 
C_  B  ->  ran  A 
C_  ran  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3121   `'ccnv 4610   dom cdm 4611   ran crn 4612
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619  df-dm 4621  df-rn 4622
This theorem is referenced by:  imass1  4986  imass2  4987  rnxpss2  5044  ssxpbm  5046  ssxp2  5048  ssrnres  5053  funssxp  5367  fssres  5373  dff2  5640  fliftf  5778  1stcof  6142  2ndcof  6143  smores  6271  tfrcllembfn  6336  caserel  7064  frecuzrdgtcl  10368  lmss  13040  txss12  13060  txbasval  13061
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