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Theorem rnss 4739
Description: Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
rnss (𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)

Proof of Theorem rnss
StepHypRef Expression
1 cnvss 4682 . . 3 (𝐴𝐵𝐴𝐵)
2 dmss 4708 . . 3 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
31, 2syl 14 . 2 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
4 df-rn 4520 . 2 ran 𝐴 = dom 𝐴
5 df-rn 4520 . 2 ran 𝐵 = dom 𝐵
63, 4, 53sstr4g 3110 1 (𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3041  ccnv 4508  dom cdm 4509  ran crn 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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by:  imass1  4884  imass2  4885  rnxpss2  4942  ssxpbm  4944  ssxp2  4946  ssrnres  4951  funssxp  5262  fssres  5268  dff2  5532  fliftf  5668  1stcof  6029  2ndcof  6030  smores  6157  tfrcllembfn  6222  caserel  6940  frecuzrdgtcl  10153  lmss  12342  txss12  12362  txbasval  12363
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