Theorem brlb 34184

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 34106	. . 3 ⊢ LB𝑅 = UB◡𝑅
2	1	breqi 5076	. 2 ⊢ (𝑆LB𝑅𝐴 ↔ 𝑆UB◡𝑅𝐴)
3		brub.1	. . 3 ⊢ 𝑆 ∈ V
4		brub.2	. . 3 ⊢ 𝐴 ∈ V
5	3, 4	brub 34183	. 2 ⊢ (𝑆UB◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴)
6		vex 3426	. . . 4 ⊢ 𝑥 ∈ V
7	6, 4	brcnv 5780	. . 3 ⊢ (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥)
8	7	ralbii 3090	. 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥)
9	2, 5, 8	3bitri 296	1 ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥)

Syntax hints:   ↔ wb 205   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   class class class wbr 5070  ^◡ccnv 5579  UBcub 34081  LBclb 34082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-eprel 5486  df-xp 5586  df-cnv 5588  df-co 5589  df-ub 34105  df-lb 34106

This theorem is referenced by: (None)