Theorem brlb 36428

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 36348	. . 3 ⊢ LB𝑅 = UB◡𝑅
2	1	breqi 5116	. 2 ⊢ (𝑆LB𝑅𝐴 ↔ 𝑆UB◡𝑅𝐴)
3		brub.1	. . 3 ⊢ 𝑆 ∈ V
4		brub.2	. . 3 ⊢ 𝐴 ∈ V
5	3, 4	brub 36427	. 2 ⊢ (𝑆UB◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴)
6		vex 3459	. . . 4 ⊢ 𝑥 ∈ V
7	6, 4	brcnv 5870	. . 3 ⊢ (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥)
8	7	ralbii 3111	. 2 ⊢ (∀𝑥 ∈ 𝑆 𝑥◡𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥)
9	2, 5, 8	3bitri 300	1 ⊢ (𝑆LB𝑅𝐴 ↔ ∀𝑥 ∈ 𝑆 𝐴𝑅𝑥)

Syntax hints:   ↔ wb 209   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   class class class wbr 5110  ^◡ccnv 5662  UBcub 36323  LBclb 36324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-eprel 5563  df-xp 5669  df-cnv 5671  df-co 5672  df-ub 36347  df-lb 36348

This theorem is referenced by: (None)