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Theorem brlb 31034
Description: Binary relationship 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 30955 . . 3 LB𝑅 = UB𝑅
21breqi 4579 . 2 (𝑆LB𝑅𝐴𝑆UB𝑅𝐴)
3 brub.1 . . 3 𝑆 ∈ V
4 brub.2 . . 3 𝐴 ∈ V
53, 4brub 31033 . 2 (𝑆UB𝑅𝐴 ↔ ∀𝑥𝑆 𝑥𝑅𝐴)
6 vex 3171 . . . 4 𝑥 ∈ V
76, 4brcnv 5211 . . 3 (𝑥𝑅𝐴𝐴𝑅𝑥)
87ralbii 2958 . 2 (∀𝑥𝑆 𝑥𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
92, 5, 83bitri 284 1 (𝑆LB𝑅𝐴 ↔ ∀𝑥𝑆 𝐴𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wcel 1975  wral 2891  Vcvv 3168   class class class wbr 4573  ccnv 5023  UBcub 30930  LBclb 30931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-eprel 4935  df-xp 5030  df-cnv 5032  df-co 5033  df-ub 30954  df-lb 30955
This theorem is referenced by: (None)
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