Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brlb Structured version   Visualization version   GIF version

Theorem brlb 35423
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 35345 . . 3 LB𝑅 = UB𝑅
21breqi 5145 . 2 (𝑆LB𝑅𝐴𝑆UB𝑅𝐴)
3 brub.1 . . 3 𝑆 ∈ V
4 brub.2 . . 3 𝐴 ∈ V
53, 4brub 35422 . 2 (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)
6 vex 3470 . . . 4 𝑥 ∈ V
76, 4brcnv 5873 . . 3 (𝑥𝑅𝐴𝐴𝑅𝑥)
87ralbii 3085 . 2 (∀𝑥𝑆 𝑥𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
92, 5, 83bitri 297 1 (𝑆LB𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2098  wral 3053  Vcvv 3466   class class class wbr 5139  ccnv 5666  UBcub 35320  LBclb 35321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-eprel 5571  df-xp 5673  df-cnv 5675  df-co 5676  df-ub 35344  df-lb 35345
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator