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Theorem brab2ga 4358
Description: The law of concretion for a binary relation. See brab2a 4336 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
brab2ga.1
brab2ga.2  R  { <. , 
>.  |  C  D  }
Assertion
Ref Expression
brab2ga  R  C  D
Distinct variable groups:   ,,   ,,   , C,   , D,   ,,
Allowed substitution hints:   (,)    R(,)

Proof of Theorem brab2ga
StepHypRef Expression
1 brab2ga.2 . . . 4  R  { <. , 
>.  |  C  D  }
2 opabssxp 4357 . . . 4  { <. ,  >.  |  C  D  }  C_  C  X.  D
31, 2eqsstri 2969 . . 3  R  C_  C  X.  D
43brel 4335 . 2  R  C  D
5 df-br 3756 . . . 4  R  <. ,  >.  R
61eleq2i 2101 . . . 4  <. ,  >.  R  <. ,  >. 
{ <. , 
>.  |  C  D  }
75, 6bitri 173 . . 3  R  <. ,  >.  { <. ,  >.  |  C  D  }
8 brab2ga.1 . . . 4
98opelopab2a 3993 . . 3  C  D  <. ,  >.  { <. ,  >.  |  C  D  }
107, 9syl5bb 181 . 2  C  D  R
114, 10biadan2 429 1  R  C  D
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390   <.cop 3370   class class class wbr 3755   {copab 3808    X. cxp 4286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294
This theorem is referenced by:  reapval  7360  ltxr  8465
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