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Theorem brresi 5855
Description: Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
Hypothesis
Ref Expression
opelresi.1 𝐶 ∈ V
Assertion
Ref Expression
brresi (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶))

Proof of Theorem brresi
StepHypRef Expression
1 opelresi.1 . 2 𝐶 ∈ V
2 brres 5853 . 2 (𝐶 ∈ V → (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶)))
31, 2ax-mp 5 1 (𝐵(𝑅𝐴)𝐶 ↔ (𝐵𝐴𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2105  Vcvv 3492   class class class wbr 5057  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-res 5560
This theorem is referenced by:  dfres2  5902  poirr2  5977  cores  6095  resco  6096  rnco  6098  fnres  6467  fvres  6682  nfunsn  6700  1stconst  7784  2ndconst  7785  fsplit  7801  fsplitOLD  7802  wfrlem5  7948  dprd2da  19093  metustid  23091  dvres  24436  dvres2  24437  ltgov  26310  hlimadd  28897  hhcmpl  28904  hhcms  28907  hlim0  28939  dfpo2  32888  eqfunresadj  32901  dfdm5  32913  dfrn5  32914  fprlem1  33034  frrlem15  33039  txpss3v  33236  brtxp  33238  pprodss4v  33242  brpprod  33243  brimg  33295  brapply  33296  funpartfun  33301  dfrdg4  33309  xrnss3v  35504  funressnfv  43155  funressnvmo  43157  afv2res  43315  setrec2lem2  44725
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