ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  brinxp2 Unicode version

Theorem brinxp2 4473
Description: Intersection of binary relation with cross product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
brinxp2  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )

Proof of Theorem brinxp2
StepHypRef Expression
1 brin 3867 . 2  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A R B  /\  A ( C  X.  D ) B ) )
2 ancom 262 . 2  |-  ( ( A R B  /\  A ( C  X.  D ) B )  <-> 
( A ( C  X.  D ) B  /\  A R B ) )
3 brxp 4441 . . . 4  |-  ( A ( C  X.  D
) B  <->  ( A  e.  C  /\  B  e.  D ) )
43anbi1i 446 . . 3  |-  ( ( A ( C  X.  D ) B  /\  A R B )  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
5 df-3an 924 . . 3  |-  ( ( A  e.  C  /\  B  e.  D  /\  A R B )  <->  ( ( A  e.  C  /\  B  e.  D )  /\  A R B ) )
64, 5bitr4i 185 . 2  |-  ( ( A ( C  X.  D ) B  /\  A R B )  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )
71, 2, 63bitri 204 1  |-  ( A ( R  i^i  ( C  X.  D ) ) B  <->  ( A  e.  C  /\  B  e.  D  /\  A R B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    /\ w3a 922    e. wcel 1436    i^i cin 2987   class class class wbr 3820    X. cxp 4409
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-xp 4417
This theorem is referenced by:  brinxp  4474  fncnv  5045  erinxp  6318
  Copyright terms: Public domain W3C validator