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Theorem brelrn 5898
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 5897 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
41, 2, 3mp3an12 1452 1 (𝐴𝐶𝐵𝐵 ∈ ran 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3444   class class class wbr 5106  ran crn 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-cnv 5642  df-dm 5644  df-rn 5645
This theorem is referenced by:  opelrn  5899  dfco2a  6199  cores  6202  dffun9  6531  funcnv  6571  rntpos  8171  rnttrcl  9663  aceq3lem  10061  axdclem  10460  axdclem2  10461  cotr2g  14867  shftfval  14961  psdmrn  18467  metustexhalf  23928  itg1addlem4  25079  itg1addlem4OLD  25080
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