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Theorem brlb 33530
 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 33452 . . 3 LB𝑅 = UB𝑅
21breqi 5039 . 2 (𝑆LB𝑅𝐴𝑆UB𝑅𝐴)
3 brub.1 . . 3 𝑆 ∈ V
4 brub.2 . . 3 𝐴 ∈ V
53, 4brub 33529 . 2 (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)
6 vex 3447 . . . 4 𝑥 ∈ V
76, 4brcnv 5721 . . 3 (𝑥𝑅𝐴𝐴𝑅𝑥)
87ralbii 3136 . 2 (∀𝑥𝑆 𝑥𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
92, 5, 83bitri 300 1 (𝑆LB𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∈ wcel 2112  ∀wral 3109  Vcvv 3444   class class class wbr 5033  ◡ccnv 5522  UBcub 33427  LBclb 33428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-eprel 5433  df-xp 5529  df-cnv 5531  df-co 5532  df-ub 33451  df-lb 33452 This theorem is referenced by: (None)
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