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

Theorem cvrfval 29967
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 2956 . 2  |-  ( K  e.  A  ->  K  e.  _V )
2 cvrfval.c . . 3  |-  C  =  (  <o  `  K )
3 fveq2 5720 . . . . . . . . 9  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
4 cvrfval.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2485 . . . . . . . 8  |-  ( p  =  K  ->  ( Base `  p )  =  B )
65eleq2d 2502 . . . . . . 7  |-  ( p  =  K  ->  (
x  e.  ( Base `  p )  <->  x  e.  B ) )
75eleq2d 2502 . . . . . . 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 5720 . . . . . . . 8  |-  ( p  =  K  ->  ( lt `  p )  =  ( lt `  K
) )
10 cvrfval.s . . . . . . . 8  |-  .<  =  ( lt `  K )
119, 10syl6eqr 2485 . . . . . . 7  |-  ( p  =  K  ->  ( lt `  p )  = 
.<  )
1211breqd 4215 . . . . . 6  |-  ( p  =  K  ->  (
x ( lt `  p ) y  <->  x  .<  y ) )
1311breqd 4215 . . . . . . . . 9  |-  ( p  =  K  ->  (
x ( lt `  p ) z  <->  x  .<  z ) )
1411breqd 4215 . . . . . . . . 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 2909 . . . . . . 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 4263 . . . 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 29965 . . . 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 4264 . . . . 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 5734 . . . . . . . 8  |-  ( Base `  K )  e.  _V
244, 23eqeltri 2505 . . . . . . 7  |-  B  e. 
_V
2524, 24xpex 4982 . . . . . 6  |-  ( B  X.  B )  e. 
_V
26 opabssxp 4942 . . . . . 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 4340 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  ( x  .<  y  /\  -.  E. z  e.  B  (
x  .<  z  /\  z  .<  y ) ) ) }  e.  _V
2822, 27eqeltri 2505 . . . 4  |-  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B
)  /\  x  .<  y  /\  -.  E. z  e.  B  ( x  .<  z  /\  z  .< 
y ) ) }  e.  _V
2919, 20, 28fvmpt 5798 . . 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 2479 . 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 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948   class class class wbr 4204   {copab 4257    X. cxp 4868   ` cfv 5446   Basecbs 13459   ltcplt 14388    <o ccvr 29961
This theorem is referenced by:  cvrval  29968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-covers 29965
  Copyright terms: Public domain W3C validator