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Theorem brub 30770
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 4884 . . . . 5  |-  ( S ( _V  X.  _V ) A  <->  ( S  e. 
_V  /\  A  e.  _V ) )
41, 2, 3mpbir2an 936 . . . 4  |-  S ( _V  X.  _V ) A
5 brdif 4467 . . . 4  |-  ( S ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) ) A  <->  ( S
( _V  X.  _V ) A  /\  -.  S
( ( _V  \  R )  o.  `'  _E  ) A ) )
64, 5mpbiran 934 . . 3  |-  ( S ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) ) A  <->  -.  S
( ( _V  \  R )  o.  `'  _E  ) A )
71, 2coepr 30441 . . 3  |-  ( S ( ( _V  \  R )  o.  `'  _E  ) A  <->  E. x  e.  S  x ( _V  \  R ) A )
86, 7xchbinx 316 . 2  |-  ( S ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) ) A  <->  -.  E. x  e.  S  x ( _V  \  R ) A )
9 df-ub 30691 . . 3  |- UB R  =  ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) )
109breqi 4422 . 2  |-  ( SUB R A  <->  S (
( _V  X.  _V )  \  ( ( _V 
\  R )  o.  `'  _E  ) ) A )
11 brv 30693 . . . . . 6  |-  x _V A
12 brdif 4467 . . . . . 6  |-  ( x ( _V  \  R
) A  <->  ( x _V A  /\  -.  x R A ) )
1311, 12mpbiran 934 . . . . 5  |-  ( x ( _V  \  R
) A  <->  -.  x R A )
1413rexbii 2901 . . . 4  |-  ( E. x  e.  S  x ( _V  \  R
) A  <->  E. x  e.  S  -.  x R A )
15 rexnal 2848 . . . 4  |-  ( E. x  e.  S  -.  x R A  <->  -.  A. x  e.  S  x R A )
1614, 15bitri 257 . . 3  |-  ( E. x  e.  S  x ( _V  \  R
) A  <->  -.  A. x  e.  S  x R A )
1716con2bii 338 . 2  |-  ( A. x  e.  S  x R A  <->  -.  E. x  e.  S  x ( _V  \  R ) A )
188, 10, 173bitr4i 285 1  |-  ( SUB R A  <->  A. x  e.  S  x R A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    e. wcel 1898   A.wral 2749   E.wrex 2750   _Vcvv 3057    \ cdif 3413   class class class wbr 4416    _E cep 4762    X. cxp 4851   `'ccnv 4852    o. ccom 4857  UBcub 30667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-eprel 4764  df-xp 4859  df-cnv 4861  df-co 4862  df-ub 30691
This theorem is referenced by:  brlb  30771
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