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Theorem brlb 33313
Description: Binary relation form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.)
Hypotheses
Ref Expression
brub.1 𝑆 ∈ V
brub.2 𝐴 ∈ V
Assertion
Ref Expression
brlb (𝑆LB𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅   𝑥,𝑆

Proof of Theorem brlb
StepHypRef Expression
1 df-lb 33235 . . 3 LB𝑅 = UB𝑅
21breqi 5063 . 2 (𝑆LB𝑅𝐴𝑆UB𝑅𝐴)
3 brub.1 . . 3 𝑆 ∈ V
4 brub.2 . . 3 𝐴 ∈ V
53, 4brub 33312 . 2 (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)
6 vex 3495 . . . 4 𝑥 ∈ V
76, 4brcnv 5746 . . 3 (𝑥𝑅𝐴𝐴𝑅𝑥)
87ralbii 3162 . 2 (∀𝑥𝑆 𝑥𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
92, 5, 83bitri 298 1 (𝑆LB𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wcel 2105  wral 3135  Vcvv 3492   class class class wbr 5057  ccnv 5547  UBcub 33210  LBclb 33211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-eprel 5458  df-xp 5554  df-cnv 5556  df-co 5557  df-ub 33234  df-lb 33235
This theorem is referenced by: (None)
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