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Theorem brub 30506
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 4885 . . . . 5  |-  ( S ( _V  X.  _V ) A  <->  ( S  e. 
_V  /\  A  e.  _V ) )
41, 2, 3mpbir2an 928 . . . 4  |-  S ( _V  X.  _V ) A
5 brdif 4476 . . . 4  |-  ( S ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) ) A  <->  ( S
( _V  X.  _V ) A  /\  -.  S
( ( _V  \  R )  o.  `'  _E  ) A ) )
64, 5mpbiran 926 . . 3  |-  ( S ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) ) A  <->  -.  S
( ( _V  \  R )  o.  `'  _E  ) A )
71, 2coepr 30179 . . 3  |-  ( S ( ( _V  \  R )  o.  `'  _E  ) A  <->  E. x  e.  S  x ( _V  \  R ) A )
86, 7xchbinx 311 . 2  |-  ( S ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) ) A  <->  -.  E. x  e.  S  x ( _V  \  R ) A )
9 df-ub 30427 . . 3  |- UB R  =  ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) )
109breqi 4432 . 2  |-  ( SUB R A  <->  S (
( _V  X.  _V )  \  ( ( _V 
\  R )  o.  `'  _E  ) ) A )
11 brv 30429 . . . . . 6  |-  x _V A
12 brdif 4476 . . . . . 6  |-  ( x ( _V  \  R
) A  <->  ( x _V A  /\  -.  x R A ) )
1311, 12mpbiran 926 . . . . 5  |-  ( x ( _V  \  R
) A  <->  -.  x R A )
1413rexbii 2934 . . . 4  |-  ( E. x  e.  S  x ( _V  \  R
) A  <->  E. x  e.  S  -.  x R A )
15 rexnal 2880 . . . 4  |-  ( E. x  e.  S  -.  x R A  <->  -.  A. x  e.  S  x R A )
1614, 15bitri 252 . . 3  |-  ( E. x  e.  S  x ( _V  \  R
) A  <->  -.  A. x  e.  S  x R A )
1716con2bii 333 . 2  |-  ( A. x  e.  S  x R A  <->  -.  E. x  e.  S  x ( _V  \  R ) A )
188, 10, 173bitr4i 280 1  |-  ( SUB R A  <->  A. x  e.  S  x R A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    e. wcel 1870   A.wral 2782   E.wrex 2783   _Vcvv 3087    \ cdif 3439   class class class wbr 4426    _E cep 4763    X. cxp 4852   `'ccnv 4853    o. ccom 4858  UBcub 30403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-eprel 4765  df-xp 4860  df-cnv 4862  df-co 4863  df-ub 30427
This theorem is referenced by:  brlb  30507
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