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Theorem brlb 30548
Description: Binary relationship form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.)
Hypotheses
Ref Expression
brub.1  |-  S  e. 
_V
brub.2  |-  A  e. 
_V
Assertion
Ref Expression
brlb  |-  ( SLB R A  <->  A. x  e.  S  A R x )
Distinct variable groups:    x, A    x, R    x, S

Proof of Theorem brlb
StepHypRef Expression
1 df-lb 30469 . . 3  |- LB R  = UB `' R
21breqi 4423 . 2  |-  ( SLB R A  <->  SUB `' R A )
3 brub.1 . . 3  |-  S  e. 
_V
4 brub.2 . . 3  |-  A  e. 
_V
53, 4brub 30547 . 2  |-  ( SUB `' R A  <->  A. x  e.  S  x `' R A )
6 vex 3081 . . . 4  |-  x  e. 
_V
76, 4brcnv 5028 . . 3  |-  ( x `' R A  <->  A R x )
87ralbii 2854 . 2  |-  ( A. x  e.  S  x `' R A  <->  A. x  e.  S  A R x )
92, 5, 83bitri 274 1  |-  ( SLB R A  <->  A. x  e.  S  A R x )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    e. wcel 1867   A.wral 2773   _Vcvv 3078   class class class wbr 4417   `'ccnv 4844  UBcub 30444  LBclb 30445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-eprel 4756  df-xp 4851  df-cnv 4853  df-co 4854  df-ub 30468  df-lb 30469
This theorem is referenced by: (None)
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