ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  brres Unicode version

Theorem brres 4707
Description: Binary relation on a restriction. (Contributed by NM, 12-Dec-2006.)
Hypothesis
Ref Expression
opelres.1  |-  B  e. 
_V
Assertion
Ref Expression
brres  |-  ( A ( C  |`  D ) B  <->  ( A C B  /\  A  e.  D ) )

Proof of Theorem brres
StepHypRef Expression
1 opelres.1 . . 3  |-  B  e. 
_V
21opelres 4706 . 2  |-  ( <. A ,  B >.  e.  ( C  |`  D )  <-> 
( <. A ,  B >.  e.  C  /\  A  e.  D ) )
3 df-br 3838 . 2  |-  ( A ( C  |`  D ) B  <->  <. A ,  B >.  e.  ( C  |`  D ) )
4 df-br 3838 . . 3  |-  ( A C B  <->  <. A ,  B >.  e.  C )
54anbi1i 446 . 2  |-  ( ( A C B  /\  A  e.  D )  <->  (
<. A ,  B >.  e.  C  /\  A  e.  D ) )
62, 3, 53bitr4i 210 1  |-  ( A ( C  |`  D ) B  <->  ( A C B  /\  A  e.  D ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    e. wcel 1438   _Vcvv 2619   <.cop 3444   class class class wbr 3837    |` cres 4430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-res 4440
This theorem is referenced by:  dfres2  4751  dfima2  4763  poirr2  4811  cores  4921  resco  4922  rnco  4924  fnres  5116  fvres  5313  nfunsn  5322  1stconst  5968  2ndconst  5969
  Copyright terms: Public domain W3C validator