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Theorem ipoval 15329
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 2986 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 ipoval.i . . 3  |-  I  =  (toInc `  F )
3 vex 2980 . . . . . . . 8  |-  f  e. 
_V
43, 3xpex 6513 . . . . . . 7  |-  ( f  X.  f )  e. 
_V
5 simpl 457 . . . . . . . . . 10  |-  ( ( { x ,  y }  C_  f  /\  x  C_  y )  ->  { x ,  y }  C_  f )
6 vex 2980 . . . . . . . . . . 11  |-  x  e. 
_V
7 vex 2980 . . . . . . . . . . 11  |-  y  e. 
_V
86, 7prss 4032 . . . . . . . . . 10  |-  ( ( x  e.  f  /\  y  e.  f )  <->  { x ,  y } 
C_  f )
95, 8sylibr 212 . . . . . . . . 9  |-  ( ( { x ,  y }  C_  f  /\  x  C_  y )  -> 
( x  e.  f  /\  y  e.  f ) )
109ssopab2i 4621 . . . . . . . 8  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) } 
C_  { <. x ,  y >.  |  ( x  e.  f  /\  y  e.  f ) }
11 df-xp 4851 . . . . . . . 8  |-  ( f  X.  f )  =  { <. x ,  y
>.  |  ( x  e.  f  /\  y  e.  f ) }
1210, 11sseqtr4i 3394 . . . . . . 7  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) } 
C_  ( f  X.  f )
134, 12ssexi 4442 . . . . . 6  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  e.  _V
1413a1i 11 . . . . 5  |-  ( f  =  F  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  e.  _V )
15 sseq2 3383 . . . . . . . 8  |-  ( f  =  F  ->  ( { x ,  y }  C_  f  <->  { x ,  y }  C_  F ) )
1615anbi1d 704 . . . . . . 7  |-  ( f  =  F  ->  (
( { x ,  y }  C_  f  /\  x  C_  y )  <-> 
( { x ,  y }  C_  F  /\  x  C_  y ) ) )
1716opabbidv 4360 . . . . . 6  |-  ( f  =  F  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  =  { <. x ,  y >.  |  ( { x ,  y }  C_  F  /\  x  C_  y ) } )
18 ipoval.l . . . . . 6  |-  .<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  F  /\  x  C_  y
) }
1917, 18syl6eqr 2493 . . . . 5  |-  ( f  =  F  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  =  .<_  )
20 simpl 457 . . . . . . . 8  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
f  =  F )
2120opeq2d 4071 . . . . . . 7  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  <. ( Base `  ndx ) ,  f >.  = 
<. ( Base `  ndx ) ,  F >. )
22 simpr 461 . . . . . . . . 9  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
o  =  .<_  )
2322fveq2d 5700 . . . . . . . 8  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
(ordTop `  o )  =  (ordTop `  .<_  ) )
2423opeq2d 4071 . . . . . . 7  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  <. (TopSet `  ndx ) ,  (ordTop `  o ) >.  =  <. (TopSet `  ndx ) ,  (ordTop `  .<_  )
>. )
2521, 24preq12d 3967 . . . . . 6  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  { <. ( Base `  ndx ) ,  f >. , 
<. (TopSet `  ndx ) ,  (ordTop `  o ) >. }  =  { <. (
Base `  ndx ) ,  F >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  .<_  ) >. } )
2622opeq2d 4071 . . . . . . 7  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  <. ( le `  ndx ) ,  o >.  = 
<. ( le `  ndx ) ,  .<_  >. )
27 id 22 . . . . . . . . . 10  |-  ( f  =  F  ->  f  =  F )
28 rabeq 2971 . . . . . . . . . . 11  |-  ( f  =  F  ->  { y  e.  f  |  ( y  i^i  x )  =  (/) }  =  {
y  e.  F  | 
( y  i^i  x
)  =  (/) } )
2928unieqd 4106 . . . . . . . . . 10  |-  ( f  =  F  ->  U. {
y  e.  f  |  ( y  i^i  x
)  =  (/) }  =  U. { y  e.  F  |  ( y  i^i  x )  =  (/) } )
3027, 29mpteq12dv 4375 . . . . . . . . 9  |-  ( f  =  F  ->  (
x  e.  f  |->  U. { y  e.  f  |  ( y  i^i  x )  =  (/) } )  =  ( x  e.  F  |->  U. {
y  e.  F  | 
( y  i^i  x
)  =  (/) } ) )
3130adantr 465 . . . . . . . 8  |-  ( ( 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 )  =  (/) } ) )
3231opeq2d 4071 . . . . . . 7  |-  ( ( 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 )  =  (/) } ) >. )
3326, 32preq12d 3967 . . . . . 6  |-  ( ( 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 )  =  (/) } ) >. } )
3425, 33uneq12d 3516 . . . . 5  |-  ( ( 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 )  =  (/) } ) >. } ) )
3514, 19, 34csbied2 3320 . . . 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 )  =  (/) } ) >. } ) )
36 df-ipo 15327 . . . 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
)  =  (/) } )
>. } ) )
37 prex 4539 . . . . 5  |-  { <. (
Base `  ndx ) ,  F >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  .<_  ) >. }  e.  _V
38 prex 4539 . . . . 5  |-  { <. ( le `  ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |-> 
U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. }  e.  _V
3937, 38unex 6383 . . . 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
4035, 36, 39fvmpt 5779 . . 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 )  =  (/) } ) >. } ) )
412, 40syl5eq 2487 . 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 )  =  (/) } ) >. } ) )
421, 41syl 16 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2724   _Vcvv 2977   [_csb 3293    u. cun 3331    i^i cin 3332    C_ wss 3333   (/)c0 3642   {cpr 3884   <.cop 3888   U.cuni 4096   {copab 4354    e. cmpt 4355    X. cxp 4843   ` cfv 5423   ndxcnx 14176   Basecbs 14179  TopSetcts 14249   lecple 14250   occoc 14251  ordTopcordt 14442  toInccipo 15326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-ipo 15327
This theorem is referenced by:  ipobas  15330  ipolerval  15331  ipotset  15332
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