Theorem brlb 35956

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 35878	. . 3 ⊢ LB𝑅 = UB◡𝑅
2	1	breqi 5149	. 2 ⊢ (𝑆LB𝑅𝐴 ↔ 𝑆UB◡𝑅𝐴)
3		brub.1	. . 3 ⊢ 𝑆 ∈ V
4		brub.2	. . 3 ⊢ 𝐴 ∈ V
5	3, 4	brub 35955	. 2 ⊢ (𝑆UB◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴)
6		vex 3484	. . . 4 ⊢ 𝑥 ∈ V
7	6, 4	brcnv 5893	. . 3 ⊢ (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥)
8	7	ralbii 3093	. 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥)
9	2, 5, 8	3bitri 297	1 ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥)

Syntax hints:   ↔ wb 206   ∈ wcel 2108  ∀wral 3061  Vcvv 3480   class class class wbr 5143  ^◡ccnv 5684  UBcub 35853  LBclb 35854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-xp 5691  df-cnv 5693  df-co 5694  df-ub 35877  df-lb 35878

This theorem is referenced by: (None)