Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrfval Unicode version

Theorem cvrfval 29458
Description: Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
cvrfval.b  |-  B  =  ( Base `  K
)
cvrfval.s  |-  .<  =  ( lt `  K )
cvrfval.c  |-  C  =  (  <o  `  K )
Assertion
Ref Expression
cvrfval  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( (
x  e.  B  /\  y  e.  B )  /\  x  .<  y  /\  -.  E. z  e.  B  ( x  .<  z  /\  z  .<  y ) ) } )
Distinct variable groups:    x, y,
z, B    x, K, y, z
Allowed substitution hints:    A( x, y, z)    C( x, y, z)    .< ( x, y, z)

Proof of Theorem cvrfval
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 cvrfval.c . . 3  |-  C  =  (  <o  `  K )
3 fveq2 5525 . . . . . . . . 9  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
4 cvrfval.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2333 . . . . . . . 8  |-  ( p  =  K  ->  ( Base `  p )  =  B )
65eleq2d 2350 . . . . . . 7  |-  ( p  =  K  ->  (
x  e.  ( Base `  p )  <->  x  e.  B ) )
75eleq2d 2350 . . . . . . 7  |-  ( p  =  K  ->  (
y  e.  ( Base `  p )  <->  y  e.  B ) )
86, 7anbi12d 691 . . . . . 6  |-  ( p  =  K  ->  (
( x  e.  (
Base `  p )  /\  y  e.  ( Base `  p ) )  <-> 
( x  e.  B  /\  y  e.  B
) ) )
9 fveq2 5525 . . . . . . . 8  |-  ( p  =  K  ->  ( lt `  p )  =  ( lt `  K
) )
10 cvrfval.s . . . . . . . 8  |-  .<  =  ( lt `  K )
119, 10syl6eqr 2333 . . . . . . 7  |-  ( p  =  K  ->  ( lt `  p )  = 
.<  )
1211breqd 4034 . . . . . 6  |-  ( p  =  K  ->  (
x ( lt `  p ) y  <->  x  .<  y ) )
1311breqd 4034 . . . . . . . . 9  |-  ( p  =  K  ->  (
x ( lt `  p ) z  <->  x  .<  z ) )
1411breqd 4034 . . . . . . . . 9  |-  ( p  =  K  ->  (
z ( lt `  p ) y  <->  z  .<  y ) )
1513, 14anbi12d 691 . . . . . . . 8  |-  ( p  =  K  ->  (
( x ( lt
`  p ) z  /\  z ( lt
`  p ) y )  <->  ( x  .<  z  /\  z  .<  y
) ) )
165, 15rexeqbidv 2749 . . . . . . 7  |-  ( p  =  K  ->  ( E. z  e.  ( Base `  p ) ( x ( lt `  p ) z  /\  z ( lt `  p ) y )  <->  E. z  e.  B  ( x  .<  z  /\  z  .<  y ) ) )
1716notbid 285 . . . . . 6  |-  ( p  =  K  ->  ( -.  E. z  e.  (
Base `  p )
( x ( lt
`  p ) z  /\  z ( lt
`  p ) y )  <->  -.  E. z  e.  B  ( x  .<  z  /\  z  .< 
y ) ) )
188, 12, 173anbi123d 1252 . . . . 5  |-  ( p  =  K  ->  (
( ( x  e.  ( Base `  p
)  /\  y  e.  ( Base `  p )
)  /\  x ( lt `  p ) y  /\  -.  E. z  e.  ( Base `  p
) ( x ( lt `  p ) z  /\  z ( lt `  p ) y ) )  <->  ( (
x  e.  B  /\  y  e.  B )  /\  x  .<  y  /\  -.  E. z  e.  B  ( x  .<  z  /\  z  .<  y ) ) ) )
1918opabbidv 4082 . . . 4  |-  ( p  =  K  ->  { <. x ,  y >.  |  ( ( x  e.  (
Base `  p )  /\  y  e.  ( Base `  p ) )  /\  x ( lt
`  p ) y  /\  -.  E. z  e.  ( Base `  p
) ( x ( lt `  p ) z  /\  z ( lt `  p ) y ) ) }  =  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  .<  y  /\  -.  E. z  e.  B  ( x  .<  z  /\  z  .< 
y ) ) } )
20 df-covers 29456 . . . 4  |-  <o  =  ( p  e.  _V  |->  {
<. x ,  y >.  |  ( ( x  e.  ( Base `  p
)  /\  y  e.  ( Base `  p )
)  /\  x ( lt `  p ) y  /\  -.  E. z  e.  ( Base `  p
) ( x ( lt `  p ) z  /\  z ( lt `  p ) y ) ) } )
21 3anass 938 . . . . . 6  |-  ( ( ( x  e.  B  /\  y  e.  B
)  /\  x  .<  y  /\  -.  E. z  e.  B  ( x  .<  z  /\  z  .< 
y ) )  <->  ( (
x  e.  B  /\  y  e.  B )  /\  ( x  .<  y  /\  -.  E. z  e.  B  ( x  .<  z  /\  z  .<  y
) ) ) )
2221opabbii 4083 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  .<  y  /\  -.  E. z  e.  B  ( x  .<  z  /\  z  .< 
y ) ) }  =  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  ( x  .<  y  /\  -.  E. z  e.  B  (
x  .<  z  /\  z  .<  y ) ) ) }
23 fvex 5539 . . . . . . . 8  |-  ( Base `  K )  e.  _V
244, 23eqeltri 2353 . . . . . . 7  |-  B  e. 
_V
2524, 24xpex 4801 . . . . . 6  |-  ( B  X.  B )  e. 
_V
26 opabssxp 4762 . . . . . 6  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  ( x  .<  y  /\  -.  E. z  e.  B  (
x  .<  z  /\  z  .<  y ) ) ) }  C_  ( B  X.  B )
2725, 26ssexi 4159 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  ( x  .<  y  /\  -.  E. z  e.  B  (
x  .<  z  /\  z  .<  y ) ) ) }  e.  _V
2822, 27eqeltri 2353 . . . 4  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  .<  y  /\  -.  E. z  e.  B  ( x  .<  z  /\  z  .< 
y ) ) }  e.  _V
2919, 20, 28fvmpt 5602 . . 3  |-  ( K  e.  _V  ->  (  <o  `  K )  =  { <. x ,  y
>.  |  ( (
x  e.  B  /\  y  e.  B )  /\  x  .<  y  /\  -.  E. z  e.  B  ( x  .<  z  /\  z  .<  y ) ) } )
302, 29syl5eq 2327 . 2  |-  ( K  e.  _V  ->  C  =  { <. x ,  y
>.  |  ( (
x  e.  B  /\  y  e.  B )  /\  x  .<  y  /\  -.  E. z  e.  B  ( x  .<  z  /\  z  .<  y ) ) } )
311, 30syl 15 1  |-  ( K  e.  A  ->  C  =  { <. x ,  y
>.  |  ( (
x  e.  B  /\  y  e.  B )  /\  x  .<  y  /\  -.  E. z  e.  B  ( x  .<  z  /\  z  .<  y ) ) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   class class class wbr 4023   {copab 4076    X. cxp 4687   ` cfv 5255   Basecbs 13148   ltcplt 14075    <o ccvr 29452
This theorem is referenced by:  cvrval  29459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-covers 29456
  Copyright terms: Public domain W3C validator