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Theorem rnss 4564
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 4508 . . 3 (𝐴𝐵𝐴𝐵)
2 dmss 4534 . . 3 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
31, 2syl 14 . 2 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
4 df-rn 4356 . 2 ran 𝐴 = dom 𝐴
5 df-rn 4356 . 2 ran 𝐵 = dom 𝐵
63, 4, 53sstr4g 2986 1 (𝐴𝐵 → ran 𝐴 ⊆ ran 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wss 2917  ccnv 4344  dom cdm 4345  ran crn 4346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356
This theorem is referenced by:  imass1  4700  imass2  4701  ssxpbm  4756  ssxp2  4758  ssrnres  4763  funssxp  5060  fssres  5066  dff2  5311  fliftf  5439  1stcof  5790  2ndcof  5791  smores  5907
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