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Theorem cvrfval 30080
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 2809 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 cvrfval.c . . 3  |-  C  =  (  <o  `  K )
3 fveq2 5541 . . . . . . . . 9  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
4 cvrfval.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2346 . . . . . . . 8  |-  ( p  =  K  ->  ( Base `  p )  =  B )
65eleq2d 2363 . . . . . . 7  |-  ( p  =  K  ->  (
x  e.  ( Base `  p )  <->  x  e.  B ) )
75eleq2d 2363 . . . . . . 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 5541 . . . . . . . 8  |-  ( p  =  K  ->  ( lt `  p )  =  ( lt `  K
) )
10 cvrfval.s . . . . . . . 8  |-  .<  =  ( lt `  K )
119, 10syl6eqr 2346 . . . . . . 7  |-  ( p  =  K  ->  ( lt `  p )  = 
.<  )
1211breqd 4050 . . . . . 6  |-  ( p  =  K  ->  (
x ( lt `  p ) y  <->  x  .<  y ) )
1311breqd 4050 . . . . . . . . 9  |-  ( p  =  K  ->  (
x ( lt `  p ) z  <->  x  .<  z ) )
1411breqd 4050 . . . . . . . . 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 2762 . . . . . . 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 4098 . . . 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 30078 . . . 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 4099 . . . . 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 5555 . . . . . . . 8  |-  ( Base `  K )  e.  _V
244, 23eqeltri 2366 . . . . . . 7  |-  B  e. 
_V
2524, 24xpex 4817 . . . . . 6  |-  ( B  X.  B )  e. 
_V
26 opabssxp 4778 . . . . . 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 4175 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  ( x  .<  y  /\  -.  E. z  e.  B  (
x  .<  z  /\  z  .<  y ) ) ) }  e.  _V
2822, 27eqeltri 2366 . . . 4  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  .<  y  /\  -.  E. z  e.  B  ( x  .<  z  /\  z  .< 
y ) ) }  e.  _V
2919, 20, 28fvmpt 5618 . . 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 2340 . 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 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   class class class wbr 4039   {copab 4092    X. cxp 4703   ` cfv 5271   Basecbs 13164   ltcplt 14091    <o ccvr 30074
This theorem is referenced by:  cvrval  30081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-covers 30078
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