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Theorem cvrfval 29751
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 2924 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 cvrfval.c . . 3  |-  C  =  (  <o  `  K )
3 fveq2 5687 . . . . . . . . 9  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
4 cvrfval.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2454 . . . . . . . 8  |-  ( p  =  K  ->  ( Base `  p )  =  B )
65eleq2d 2471 . . . . . . 7  |-  ( p  =  K  ->  (
x  e.  ( Base `  p )  <->  x  e.  B ) )
75eleq2d 2471 . . . . . . 7  |-  ( p  =  K  ->  (
y  e.  ( Base `  p )  <->  y  e.  B ) )
86, 7anbi12d 692 . . . . . 6  |-  ( p  =  K  ->  (
( x  e.  (
Base `  p )  /\  y  e.  ( Base `  p ) )  <-> 
( x  e.  B  /\  y  e.  B
) ) )
9 fveq2 5687 . . . . . . . 8  |-  ( p  =  K  ->  ( lt `  p )  =  ( lt `  K
) )
10 cvrfval.s . . . . . . . 8  |-  .<  =  ( lt `  K )
119, 10syl6eqr 2454 . . . . . . 7  |-  ( p  =  K  ->  ( lt `  p )  = 
.<  )
1211breqd 4183 . . . . . 6  |-  ( p  =  K  ->  (
x ( lt `  p ) y  <->  x  .<  y ) )
1311breqd 4183 . . . . . . . . 9  |-  ( p  =  K  ->  (
x ( lt `  p ) z  <->  x  .<  z ) )
1411breqd 4183 . . . . . . . . 9  |-  ( p  =  K  ->  (
z ( lt `  p ) y  <->  z  .<  y ) )
1513, 14anbi12d 692 . . . . . . . 8  |-  ( p  =  K  ->  (
( x ( lt
`  p ) z  /\  z ( lt
`  p ) y )  <->  ( x  .<  z  /\  z  .<  y
) ) )
165, 15rexeqbidv 2877 . . . . . . 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 286 . . . . . 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 1254 . . . . 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 4231 . . . 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 29749 . . . 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 940 . . . . . 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 4232 . . . . 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 5701 . . . . . . . 8  |-  ( Base `  K )  e.  _V
244, 23eqeltri 2474 . . . . . . 7  |-  B  e. 
_V
2524, 24xpex 4949 . . . . . 6  |-  ( B  X.  B )  e. 
_V
26 opabssxp 4909 . . . . . 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 4308 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  ( x  .<  y  /\  -.  E. z  e.  B  (
x  .<  z  /\  z  .<  y ) ) ) }  e.  _V
2822, 27eqeltri 2474 . . . 4  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  .<  y  /\  -.  E. z  e.  B  ( x  .<  z  /\  z  .< 
y ) ) }  e.  _V
2919, 20, 28fvmpt 5765 . . 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 2448 . 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 16 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2667   _Vcvv 2916   class class class wbr 4172   {copab 4225    X. cxp 4835   ` cfv 5413   Basecbs 13424   ltcplt 14353    <o ccvr 29745
This theorem is referenced by:  cvrval  29752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-covers 29749
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