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Theorem brresi 5830
 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 5828 . 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 209   ∧ wa 399   ∈ wcel 2111  Vcvv 3441   class class class wbr 5033   ↾ cres 5524 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3885  df-un 3887  df-in 3889  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-br 5034  df-opab 5096  df-xp 5528  df-res 5534 This theorem is referenced by:  dfres2  5879  poirr2  5954  cores  6074  resco  6075  rnco  6077  fnres  6451  fvres  6671  nfunsn  6689  1stconst  7788  2ndconst  7789  fsplit  7805  fsplitOLD  7806  wfrlem5  7957  dprd2da  19175  metustid  23199  dvres  24552  dvres2  24553  ltgov  26432  hlimadd  29017  hhcmpl  29024  hhcms  29027  hlim0  29059  dfpo2  33167  eqfunresadj  33180  dfdm5  33192  dfrn5  33193  fprlem1  33313  frrlem15  33318  txpss3v  33515  brtxp  33517  pprodss4v  33521  brpprod  33522  brimg  33574  brapply  33575  funpartfun  33580  dfrdg4  33588  xrnss3v  35851  funressnfv  43719  funressnvmo  43721  afv2res  43879  setrec2lem2  45306
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