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Theorem rnss 4613
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 4557 . . 3 (𝐴𝐵𝐴𝐵)
2 dmss 4583 . . 3 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
31, 2syl 14 . 2 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
4 df-rn 4403 . 2 ran 𝐴 = dom 𝐴
5 df-rn 4403 . 2 ran 𝐵 = dom 𝐵
63, 4, 53sstr4g 3050 1 (𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wss 2983  ccnv 4391  dom cdm 4392  ran crn 4393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-cnv 4400  df-dm 4402  df-rn 4403
This theorem is referenced by:  imass1  4751  imass2  4752  ssxpbm  4807  ssxp2  4809  ssrnres  4814  funssxp  5112  fssres  5118  dff2  5364  fliftf  5491  1stcof  5842  2ndcof  5843  smores  5962  tfrcllembfn  6027  frecuzrdgtcl  9530
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