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Theorem ipoval 14257
Description: Value of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Hypotheses
Ref Expression
ipoval.i  |-  I  =  (toInc `  F )
ipoval.l  |-  .<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  F  /\  x  C_  y
) }
Assertion
Ref Expression
ipoval  |-  ( F  e.  V  ->  I  =  ( { <. (
Base `  ndx ) ,  F >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  .<_  ) >. }  u.  {
<. ( le `  ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |->  U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. } ) )
Distinct variable groups:    x, y, F    x, I, y    x, V, y
Allowed substitution hints:    .<_ ( x, y)

Proof of Theorem ipoval
Dummy variables  f 
o are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 ipoval.i . . 3  |-  I  =  (toInc `  F )
3 vex 2791 . . . . . . . 8  |-  f  e. 
_V
43, 3xpex 4801 . . . . . . 7  |-  ( f  X.  f )  e. 
_V
5 simpl 443 . . . . . . . . . 10  |-  ( ( { x ,  y }  C_  f  /\  x  C_  y )  ->  { x ,  y }  C_  f )
6 vex 2791 . . . . . . . . . . 11  |-  x  e. 
_V
7 vex 2791 . . . . . . . . . . 11  |-  y  e. 
_V
86, 7prss 3769 . . . . . . . . . 10  |-  ( ( x  e.  f  /\  y  e.  f )  <->  { x ,  y } 
C_  f )
95, 8sylibr 203 . . . . . . . . 9  |-  ( ( { x ,  y }  C_  f  /\  x  C_  y )  -> 
( x  e.  f  /\  y  e.  f ) )
109ssopab2i 4292 . . . . . . . 8  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) } 
C_  { <. x ,  y >.  |  ( x  e.  f  /\  y  e.  f ) }
11 df-xp 4695 . . . . . . . 8  |-  ( f  X.  f )  =  { <. x ,  y
>.  |  ( x  e.  f  /\  y  e.  f ) }
1210, 11sseqtr4i 3211 . . . . . . 7  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) } 
C_  ( f  X.  f )
134, 12ssexi 4159 . . . . . 6  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  e.  _V
1413a1i 10 . . . . 5  |-  ( f  =  F  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  e.  _V )
15 sseq2 3200 . . . . . . . . . . 11  |-  ( f  =  F  ->  ( { x ,  y }  C_  f  <->  { x ,  y }  C_  F ) )
1615anbi1d 685 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( { x ,  y }  C_  f  /\  x  C_  y )  <-> 
( { x ,  y }  C_  F  /\  x  C_  y ) ) )
1716opabbidv 4082 . . . . . . . . 9  |-  ( f  =  F  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  =  { <. x ,  y >.  |  ( { x ,  y }  C_  F  /\  x  C_  y ) } )
18 ipoval.l . . . . . . . . 9  |-  .<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  F  /\  x  C_  y
) }
1917, 18syl6eqr 2333 . . . . . . . 8  |-  ( f  =  F  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  =  .<_  )
2019eqeq2d 2294 . . . . . . 7  |-  ( f  =  F  ->  (
o  =  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  <-> 
o  =  .<_  ) )
2120biimpa 470 . . . . . 6  |-  ( ( f  =  F  /\  o  =  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) } )  ->  o  =  .<_  )
22 simpl 443 . . . . . . . . 9  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
f  =  F )
2322opeq2d 3803 . . . . . . . 8  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  <. ( Base `  ndx ) ,  f >.  = 
<. ( Base `  ndx ) ,  F >. )
24 simpr 447 . . . . . . . . . 10  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
o  =  .<_  )
2524fveq2d 5529 . . . . . . . . 9  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
(ordTop `  o )  =  (ordTop `  .<_  ) )
2625opeq2d 3803 . . . . . . . 8  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  <. (TopSet `  ndx ) ,  (ordTop `  o ) >.  =  <. (TopSet `  ndx ) ,  (ordTop `  .<_  )
>. )
2723, 26preq12d 3714 . . . . . . 7  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  { <. ( Base `  ndx ) ,  f >. , 
<. (TopSet `  ndx ) ,  (ordTop `  o ) >. }  =  { <. (
Base `  ndx ) ,  F >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  .<_  ) >. } )
2824opeq2d 3803 . . . . . . . 8  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  <. ( le `  ndx ) ,  o >.  = 
<. ( le `  ndx ) ,  .<_  >. )
29 id 19 . . . . . . . . . . 11  |-  ( f  =  F  ->  f  =  F )
30 rabeq 2782 . . . . . . . . . . . 12  |-  ( f  =  F  ->  { y  e.  f  |  ( y  i^i  x )  =  (/) }  =  {
y  e.  F  | 
( y  i^i  x
)  =  (/) } )
3130unieqd 3838 . . . . . . . . . . 11  |-  ( f  =  F  ->  U. {
y  e.  f  |  ( y  i^i  x
)  =  (/) }  =  U. { y  e.  F  |  ( y  i^i  x )  =  (/) } )
3229, 31mpteq12dv 4098 . . . . . . . . . 10  |-  ( f  =  F  ->  (
x  e.  f  |->  U. { y  e.  f  |  ( y  i^i  x )  =  (/) } )  =  ( x  e.  F  |->  U. {
y  e.  F  | 
( y  i^i  x
)  =  (/) } ) )
3332adantr 451 . . . . . . . . 9  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
( x  e.  f 
|->  U. { y  e.  f  |  ( y  i^i  x )  =  (/) } )  =  ( x  e.  F  |->  U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) )
3433opeq2d 3803 . . . . . . . 8  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  <. ( oc `  ndx ) ,  ( x  e.  f  |->  U. {
y  e.  f  |  ( y  i^i  x
)  =  (/) } )
>.  =  <. ( oc
`  ndx ) ,  ( x  e.  F  |->  U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. )
3528, 34preq12d 3714 . . . . . . 7  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  { <. ( le `  ndx ) ,  o >. ,  <. ( oc `  ndx ) ,  ( x  e.  f  |->  U. {
y  e.  f  |  ( y  i^i  x
)  =  (/) } )
>. }  =  { <. ( le `  ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |-> 
U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. } )
3627, 35uneq12d 3330 . . . . . 6  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
( { <. ( Base `  ndx ) ,  f >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  o ) >. }  u.  {
<. ( le `  ndx ) ,  o >. , 
<. ( oc `  ndx ) ,  ( x  e.  f  |->  U. {
y  e.  f  |  ( y  i^i  x
)  =  (/) } )
>. } )  =  ( { <. ( Base `  ndx ) ,  F >. , 
<. (TopSet `  ndx ) ,  (ordTop `  .<_  ) >. }  u.  { <. ( le `  ndx ) , 
.<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |-> 
U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. } ) )
3721, 36syldan 456 . . . . 5  |-  ( ( f  =  F  /\  o  =  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) } )  ->  ( { <. ( Base `  ndx ) ,  f >. , 
<. (TopSet `  ndx ) ,  (ordTop `  o ) >. }  u.  { <. ( le `  ndx ) ,  o >. ,  <. ( oc `  ndx ) ,  ( x  e.  f  |->  U. { y  e.  f  |  ( y  i^i  x )  =  (/) } ) >. } )  =  ( { <. (
Base `  ndx ) ,  F >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  .<_  ) >. }  u.  {
<. ( le `  ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |->  U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. } ) )
3814, 37csbied 3123 . . . 4  |-  ( f  =  F  ->  [_ { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y
) }  /  o ]_ ( { <. ( Base `  ndx ) ,  f >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  o ) >. }  u.  {
<. ( le `  ndx ) ,  o >. , 
<. ( oc `  ndx ) ,  ( x  e.  f  |->  U. {
y  e.  f  |  ( y  i^i  x
)  =  (/) } )
>. } )  =  ( { <. ( Base `  ndx ) ,  F >. , 
<. (TopSet `  ndx ) ,  (ordTop `  .<_  ) >. }  u.  { <. ( le `  ndx ) , 
.<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |-> 
U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. } ) )
39 df-ipo 14255 . . . 4  |- toInc  =  ( f  e.  _V  |->  [_ { <. x ,  y
>.  |  ( {
x ,  y } 
C_  f  /\  x  C_  y ) }  / 
o ]_ ( { <. (
Base `  ndx ) ,  f >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  o ) >. }  u.  {
<. ( le `  ndx ) ,  o >. , 
<. ( oc `  ndx ) ,  ( x  e.  f  |->  U. {
y  e.  f  |  ( y  i^i  x
)  =  (/) } )
>. } ) )
40 prex 4217 . . . . 5  |-  { <. (
Base `  ndx ) ,  F >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  .<_  ) >. }  e.  _V
41 prex 4217 . . . . 5  |-  { <. ( le `  ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |-> 
U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. }  e.  _V
4240, 41unex 4518 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  F >. , 
<. (TopSet `  ndx ) ,  (ordTop `  .<_  ) >. }  u.  { <. ( le `  ndx ) , 
.<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |-> 
U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. } )  e.  _V
4338, 39, 42fvmpt 5602 . . 3  |-  ( F  e.  _V  ->  (toInc `  F )  =  ( { <. ( Base `  ndx ) ,  F >. , 
<. (TopSet `  ndx ) ,  (ordTop `  .<_  ) >. }  u.  { <. ( le `  ndx ) , 
.<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |-> 
U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. } ) )
442, 43syl5eq 2327 . 2  |-  ( F  e.  _V  ->  I  =  ( { <. (
Base `  ndx ) ,  F >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  .<_  ) >. }  u.  {
<. ( le `  ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |->  U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. } ) )
451, 44syl 15 1  |-  ( F  e.  V  ->  I  =  ( { <. (
Base `  ndx ) ,  F >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  .<_  ) >. }  u.  {
<. ( le `  ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |->  U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   [_csb 3081    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {cpr 3641   <.cop 3643   U.cuni 3827   {copab 4076    e. cmpt 4077    X. cxp 4687   ` cfv 5255   ndxcnx 13145   Basecbs 13148  TopSetcts 13214   lecple 13215   occoc 13216  ordTopcordt 13398  toInccipo 14254
This theorem is referenced by:  ipobas  14258  ipolerval  14259  ipotset  14260
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-csb 3082  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-ipo 14255
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