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Theorem brub 29209
Description: Binary relationship form of the upper 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
brub  |-  ( SUB R A  <->  A. x  e.  S  x R A )
Distinct variable groups:    x, A    x, R    x, S

Proof of Theorem brub
StepHypRef Expression
1 brub.1 . . . . 5  |-  S  e. 
_V
2 brub.2 . . . . 5  |-  A  e. 
_V
3 brxp 5030 . . . . 5  |-  ( S ( _V  X.  _V ) A  <->  ( S  e. 
_V  /\  A  e.  _V ) )
41, 2, 3mpbir2an 918 . . . 4  |-  S ( _V  X.  _V ) A
5 brdif 4497 . . . 4  |-  ( S ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) ) A  <->  ( S
( _V  X.  _V ) A  /\  -.  S
( ( _V  \  R )  o.  `'  _E  ) A ) )
64, 5mpbiran 916 . . 3  |-  ( S ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) ) A  <->  -.  S
( ( _V  \  R )  o.  `'  _E  ) A )
71, 2coepr 28786 . . 3  |-  ( S ( ( _V  \  R )  o.  `'  _E  ) A  <->  E. x  e.  S  x ( _V  \  R ) A )
86, 7xchbinx 310 . 2  |-  ( S ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) ) A  <->  -.  E. x  e.  S  x ( _V  \  R ) A )
9 df-ub 29130 . . 3  |- UB R  =  ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) )
109breqi 4453 . 2  |-  ( SUB R A  <->  S (
( _V  X.  _V )  \  ( ( _V 
\  R )  o.  `'  _E  ) ) A )
11 brv 29132 . . . . . 6  |-  x _V A
12 brdif 4497 . . . . . 6  |-  ( x ( _V  \  R
) A  <->  ( x _V A  /\  -.  x R A ) )
1311, 12mpbiran 916 . . . . 5  |-  ( x ( _V  \  R
) A  <->  -.  x R A )
1413rexbii 2965 . . . 4  |-  ( E. x  e.  S  x ( _V  \  R
) A  <->  E. x  e.  S  -.  x R A )
15 rexnal 2912 . . . 4  |-  ( E. x  e.  S  -.  x R A  <->  -.  A. x  e.  S  x R A )
1614, 15bitri 249 . . 3  |-  ( E. x  e.  S  x ( _V  \  R
) A  <->  -.  A. x  e.  S  x R A )
1716con2bii 332 . 2  |-  ( A. x  e.  S  x R A  <->  -.  E. x  e.  S  x ( _V  \  R ) A )
188, 10, 173bitr4i 277 1  |-  ( SUB R A  <->  A. x  e.  S  x R A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    \ cdif 3473   class class class wbr 4447    _E cep 4789    X. cxp 4997   `'ccnv 4998    o. ccom 5003  UBcub 29106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-eprel 4791  df-xp 5005  df-cnv 5007  df-co 5008  df-ub 29130
This theorem is referenced by:  brlb  29210
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