Theorem brlb 36190

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 36110	. . 3 ⊢ LB𝑅 = UB◡𝑅
2	1	breqi 5085	. 2 ⊢ (𝑆LB𝑅𝐴 ↔ 𝑆UB◡𝑅𝐴)
3		brub.1	. . 3 ⊢ 𝑆 ∈ V
4		brub.2	. . 3 ⊢ 𝐴 ∈ V
5	3, 4	brub 36189	. 2 ⊢ (𝑆UB◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴)
6		vex 3436	. . . 4 ⊢ 𝑥 ∈ V
7	6, 4	brcnv 5831	. . 3 ⊢ (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥)
8	7	ralbii 3086	. 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥)
9	2, 5, 8	3bitri 298	1 ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥)

Syntax hints:   ↔ wb 207   ∈ wcel 2119  ∀wral 3054  Vcvv 3432   class class class wbr 5079  ^◡ccnv 5624  UBcub 36085  LBclb 36086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-eprel 5525  df-xp 5631  df-cnv 5633  df-co 5634  df-ub 36109  df-lb 36110

This theorem is referenced by: (None)