Theorem brlb 32062

Description: Binary relation form of the lower bound functor. (Contributed by Scott Fenton, 3-May-2018.)

Hypotheses

Ref	Expression
brub.1	|- S e. _V
brub.2	|- A e. _V

Assertion

Ref	Expression
brlb	|- ( SLB R A <-> A. x e. S A R x )

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

Proof of Theorem brlb

Step	Hyp	Ref	Expression
1		df-lb 31984	. . 3 |- LB R = UB `' R
2	1	breqi 4659	. 2 |- ( SLB R A <-> SUB `' R A )
3		brub.1	. . 3 |- S e. _V
4		brub.2	. . 3 |- A e. _V
5	3, 4	brub 32061	. 2 |- ( SUB `' R A <-> A. x e. S x `' R A )
6		vex 3203	. . . 4 |- x e. _V
7	6, 4	brcnv 5305	. . 3 |- ( x `' R A <-> A R x )
8	7	ralbii 2980	. 2 |- ( A. x e. S x `' R A <-> A. x e. S A R x )
9	2, 5, 8	3bitri 286	1 |- ( SLB R A <-> A. x e. S A R x )

Syntax hints:   <-> wb 196   e. wcel 1990   A.wral 2912   _Vcvv 3200   class class class wbr 4653   `'ccnv 5113  UBcub 31959  LBclb 31960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-xp 5120  df-cnv 5122  df-co 5123  df-ub 31983  df-lb 31984

This theorem is referenced by: (None)