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Theorem brelrn 4681
Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)
Hypotheses
Ref Expression
brelrn.1 𝐴 ∈ V
brelrn.2 𝐵 ∈ V
Assertion
Ref Expression
brelrn (𝐴𝐶𝐵𝐵 ∈ ran 𝐶)

Proof of Theorem brelrn
StepHypRef Expression
1 brelrn.1 . 2 𝐴 ∈ V
2 brelrn.2 . 2 𝐵 ∈ V
3 brelrng 4679 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
41, 2, 3mp3an12 1264 1 (𝐴𝐶𝐵𝐵 ∈ ran 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1439  Vcvv 2620   class class class wbr 3851  ran crn 4453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-cnv 4460  df-dm 4462  df-rn 4463
This theorem is referenced by:  opelrn  4682  dfco2a  4944  cores  4947  dffun9  5057  funcnv  5088  rntpos  6036  tfrexlem  6113
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