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Theorem brlb 34586
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 34508 . . 3 LB𝑅 = UB𝑅
21breqi 5112 . 2 (𝑆LB𝑅𝐴𝑆UB𝑅𝐴)
3 brub.1 . . 3 𝑆 ∈ V
4 brub.2 . . 3 𝐴 ∈ V
53, 4brub 34585 . 2 (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)
6 vex 3448 . . . 4 𝑥 ∈ V
76, 4brcnv 5839 . . 3 (𝑥𝑅𝐴𝐴𝑅𝑥)
87ralbii 3093 . 2 (∀𝑥𝑆 𝑥𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
92, 5, 83bitri 297 1 (𝑆LB𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2107  wral 3061  Vcvv 3444   class class class wbr 5106  ccnv 5633  UBcub 34483  LBclb 34484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-eprel 5538  df-xp 5640  df-cnv 5642  df-co 5643  df-ub 34507  df-lb 34508
This theorem is referenced by: (None)
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