Theorem brlb 35995

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 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅   𝑥,𝑆

Proof of Theorem brlb

Step	Hyp	Ref	Expression
1		df-lb 35917	. . 3 ⊢ LB𝑅 = UB◡𝑅
2	1	breqi 5097	. 2 ⊢ (𝑆LB𝑅𝐴 ↔ 𝑆UB◡𝑅𝐴)
3		brub.1	. . 3 ⊢ 𝑆 ∈ V
4		brub.2	. . 3 ⊢ 𝐴 ∈ V
5	3, 4	brub 35994	. 2 ⊢ (𝑆UB◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴)
6		vex 3440	. . . 4 ⊢ 𝑥 ∈ V
7	6, 4	brcnv 5822	. . 3 ⊢ (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥)
8	7	ralbii 3078	. 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥)
9	2, 5, 8	3bitri 297	1 ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥)

Syntax hints:   ↔ wb 206   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   class class class wbr 5091  ^◡ccnv 5615  UBcub 35892  LBclb 35893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-eprel 5516  df-xp 5622  df-cnv 5624  df-co 5625  df-ub 35916  df-lb 35917

This theorem is referenced by: (None)