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Theorem brresi 5994
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 5992 . 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 205  wa 394  wcel 2098  Vcvv 3461   class class class wbr 5149  cres 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-res 5690
This theorem is referenced by:  dfres2  6046  poirr2  6131  cores  6255  resco  6256  rnco  6258  dfpo2  6302  fnres  6683  fvres  6915  nfunsn  6938  eqfunresadj  7367  1stconst  8105  2ndconst  8106  fsplit  8122  fprlem1  8306  wfrlem5OLD  8334  ttrclresv  9742  ttrclselem2  9751  frrlem15  9782  dprd2da  20011  metustid  24507  dvres  25884  dvres2  25885  ltgov  28473  hlimadd  31075  hhcmpl  31082  hhcms  31085  hlim0  31117  dfdm5  35499  dfrn5  35500  txpss3v  35605  brtxp  35607  pprodss4v  35611  brpprod  35612  brimg  35664  brapply  35665  funpartfun  35670  dfrdg4  35678  xrnss3v  37974  funressnfv  46563  funressnvmo  46565  afv2res  46757  setrec2lem2  48311
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