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Theorem brelrn 5956
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 5955 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
41, 2, 3mp3an12 1450 1 (𝐴𝐶𝐵𝐵 ∈ ran 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3478   class class class wbr 5148  ran crn 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700
This theorem is referenced by:  opelrn  5957  dfco2a  6268  cores  6271  dffun9  6597  funcnv  6637  rntpos  8263  rnttrcl  9760  aceq3lem  10158  axdclem  10557  axdclem2  10558  cotr2g  15012  shftfval  15106  psdmrn  18631  metustexhalf  24585  itg1addlem4  25748  itg1addlem4OLD  25749
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