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Theorem rncoss 4624
Description: Range of a composition. (Contributed by NM, 19-Mar-1998.)
Assertion
Ref Expression
rncoss ran (𝐴𝐵) ⊆ ran 𝐴

Proof of Theorem rncoss
StepHypRef Expression
1 dmcoss 4623 . 2 dom (𝐵𝐴) ⊆ dom 𝐴
2 df-rn 4376 . . 3 ran (𝐴𝐵) = dom (𝐴𝐵)
3 cnvco 4542 . . . 4 (𝐴𝐵) = (𝐵𝐴)
43dmeqi 4558 . . 3 dom (𝐴𝐵) = dom (𝐵𝐴)
52, 4eqtri 2102 . 2 ran (𝐴𝐵) = dom (𝐵𝐴)
6 df-rn 4376 . 2 ran 𝐴 = dom 𝐴
71, 5, 63sstr4i 3039 1 ran (𝐴𝐵) ⊆ ran 𝐴
Colors of variables: wff set class
Syntax hints:  wss 2974  ccnv 4364  dom cdm 4365  ran crn 4366  ccom 4369
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-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376
This theorem is referenced by:  cossxp  4867  fco  5081
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