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Theorem ipoval 14582
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 2966 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 ipoval.i . . 3  |-  I  =  (toInc `  F )
3 vex 2961 . . . . . . . 8  |-  f  e. 
_V
43, 3xpex 4992 . . . . . . 7  |-  ( f  X.  f )  e. 
_V
5 simpl 445 . . . . . . . . . 10  |-  ( ( { x ,  y }  C_  f  /\  x  C_  y )  ->  { x ,  y }  C_  f )
6 vex 2961 . . . . . . . . . . 11  |-  x  e. 
_V
7 vex 2961 . . . . . . . . . . 11  |-  y  e. 
_V
86, 7prss 3954 . . . . . . . . . 10  |-  ( ( x  e.  f  /\  y  e.  f )  <->  { x ,  y } 
C_  f )
95, 8sylibr 205 . . . . . . . . 9  |-  ( ( { x ,  y }  C_  f  /\  x  C_  y )  -> 
( x  e.  f  /\  y  e.  f ) )
109ssopab2i 4484 . . . . . . . 8  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) } 
C_  { <. x ,  y >.  |  ( x  e.  f  /\  y  e.  f ) }
11 df-xp 4886 . . . . . . . 8  |-  ( f  X.  f )  =  { <. x ,  y
>.  |  ( x  e.  f  /\  y  e.  f ) }
1210, 11sseqtr4i 3383 . . . . . . 7  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) } 
C_  ( f  X.  f )
134, 12ssexi 4350 . . . . . 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 3372 . . . . . . . 8  |-  ( f  =  F  ->  ( { x ,  y }  C_  f  <->  { x ,  y }  C_  F ) )
1615anbi1d 687 . . . . . . 7  |-  ( f  =  F  ->  (
( { x ,  y }  C_  f  /\  x  C_  y )  <-> 
( { x ,  y }  C_  F  /\  x  C_  y ) ) )
1716opabbidv 4273 . . . . . 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 2488 . . . . 5  |-  ( f  =  F  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  =  .<_  )
20 simpl 445 . . . . . . . 8  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
f  =  F )
2120opeq2d 3993 . . . . . . 7  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  <. ( Base `  ndx ) ,  f >.  = 
<. ( Base `  ndx ) ,  F >. )
22 simpr 449 . . . . . . . . 9  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
o  =  .<_  )
2322fveq2d 5734 . . . . . . . 8  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
(ordTop `  o )  =  (ordTop `  .<_  ) )
2423opeq2d 3993 . . . . . . 7  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  <. (TopSet `  ndx ) ,  (ordTop `  o ) >.  =  <. (TopSet `  ndx ) ,  (ordTop `  .<_  )
>. )
2521, 24preq12d 3893 . . . . . 6  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  { <. ( Base `  ndx ) ,  f >. , 
<. (TopSet `  ndx ) ,  (ordTop `  o ) >. }  =  { <. (
Base `  ndx ) ,  F >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  .<_  ) >. } )
2622opeq2d 3993 . . . . . . 7  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  <. ( le `  ndx ) ,  o >.  = 
<. ( le `  ndx ) ,  .<_  >. )
27 id 21 . . . . . . . . . 10  |-  ( f  =  F  ->  f  =  F )
28 rabeq 2952 . . . . . . . . . . 11  |-  ( f  =  F  ->  { y  e.  f  |  ( y  i^i  x )  =  (/) }  =  {
y  e.  F  | 
( y  i^i  x
)  =  (/) } )
2928unieqd 4028 . . . . . . . . . 10  |-  ( f  =  F  ->  U. {
y  e.  f  |  ( y  i^i  x
)  =  (/) }  =  U. { y  e.  F  |  ( y  i^i  x )  =  (/) } )
3027, 29mpteq12dv 4289 . . . . . . . . 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 453 . . . . . . . 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 3993 . . . . . . 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 3893 . . . . . 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 3504 . . . . 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 3296 . . . 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 14580 . . . 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 4408 . . . . 5  |-  { <. (
Base `  ndx ) ,  F >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  .<_  ) >. }  e.  _V
38 prex 4408 . . . . 5  |-  { <. ( le `  ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |-> 
U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. }  e.  _V
3937, 38unex 4709 . . . 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 5808 . . 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 2482 . 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958   [_csb 3253    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   {cpr 3817   <.cop 3819   U.cuni 4017   {copab 4267    e. cmpt 4268    X. cxp 4878   ` cfv 5456   ndxcnx 13468   Basecbs 13471  TopSetcts 13537   lecple 13538   occoc 13539  ordTopcordt 13723  toInccipo 14579
This theorem is referenced by:  ipobas  14583  ipolerval  14584  ipotset  14585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ipo 14580
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