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

Theorem rnss 4859
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 4802 . . 3 (𝐴𝐵𝐴𝐵)
2 dmss 4828 . . 3 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
31, 2syl 14 . 2 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
4 df-rn 4639 . 2 ran 𝐴 = dom 𝐴
5 df-rn 4639 . 2 ran 𝐵 = dom 𝐵
63, 4, 53sstr4g 3200 1 (𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3131  ccnv 4627  dom cdm 4628  ran crn 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-cnv 4636  df-dm 4638  df-rn 4639
This theorem is referenced by:  imass1  5005  imass2  5006  rnxpss2  5064  ssxpbm  5066  ssxp2  5068  ssrnres  5073  funssxp  5387  fssres  5393  dff2  5662  fliftf  5802  1stcof  6166  2ndcof  6167  smores  6295  tfrcllembfn  6360  caserel  7088  frecuzrdgtcl  10414  lmss  13785  txss12  13805  txbasval  13806
  Copyright terms: Public domain W3C validator