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Theorem brelrn 5560
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 5559 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
41, 2, 3mp3an12 1576 1 (𝐴𝐶𝐵𝐵 ∈ ran 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  Vcvv 3385   class class class wbr 4843  ran crn 5313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-cnv 5320  df-dm 5322  df-rn 5323
This theorem is referenced by:  opelrn  5561  dfco2a  5854  cores  5857  dffun9  6130  funcnv  6169  rntpos  7603  aceq3lem  9229  axdclem  9629  axdclem2  9630  cotr2g  14058  shftfval  14151  psdmrn  17522  metustexhalf  22689  itg1addlem4  23807
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