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Theorem brresi 5705
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 5703 . 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 198  wa 387  wcel 2050  Vcvv 3415   class class class wbr 4930  cres 5410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-br 4931  df-opab 4993  df-xp 5414  df-res 5420
This theorem is referenced by:  dfres2  5756  poirr2  5826  cores  5943  resco  5944  rnco  5946  fnres  6308  fvres  6520  nfunsn  6539  1stconst  7605  2ndconst  7606  fsplit  7622  wfrlem5  7765  dprd2da  18917  metustid  22870  dvres  24215  dvres2  24216  ltgov  26088  hlimadd  28752  hhcmpl  28759  hhcms  28762  hlim0  28794  dfpo2  32511  eqfunresadj  32524  dfdm5  32536  dfrn5  32537  fprlem1  32658  frrlem15  32663  txpss3v  32860  brtxp  32862  pprodss4v  32866  brpprod  32867  brimg  32919  brapply  32920  funpartfun  32925  dfrdg4  32933  xrnss3v  35069  funressnfv  42684  funressnvmo  42686  afv2res  42845  setrec2lem2  44165
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