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Theorem ipoval 15910
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 3118 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 ipoval.i . . 3  |-  I  =  (toInc `  F )
3 vex 3112 . . . . . . . 8  |-  f  e. 
_V
43, 3xpex 6603 . . . . . . 7  |-  ( f  X.  f )  e. 
_V
5 simpl 457 . . . . . . . . . 10  |-  ( ( { x ,  y }  C_  f  /\  x  C_  y )  ->  { x ,  y }  C_  f )
6 vex 3112 . . . . . . . . . . 11  |-  x  e. 
_V
7 vex 3112 . . . . . . . . . . 11  |-  y  e. 
_V
86, 7prss 4186 . . . . . . . . . 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 4784 . . . . . . . 8  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) } 
C_  { <. x ,  y >.  |  ( x  e.  f  /\  y  e.  f ) }
11 df-xp 5014 . . . . . . . 8  |-  ( f  X.  f )  =  { <. x ,  y
>.  |  ( x  e.  f  /\  y  e.  f ) }
1210, 11sseqtr4i 3532 . . . . . . 7  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) } 
C_  ( f  X.  f )
134, 12ssexi 4601 . . . . . 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 3521 . . . . . . . 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 4520 . . . . . 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 2516 . . . . 5  |-  ( f  =  F  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  =  .<_  )
20 simpl 457 . . . . . . . 8  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
f  =  F )
2120opeq2d 4226 . . . . . . 7  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  <. ( Base `  ndx ) ,  f >.  = 
<. ( Base `  ndx ) ,  F >. )
22 simpr 461 . . . . . . . . 9  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
o  =  .<_  )
2322fveq2d 5876 . . . . . . . 8  |-  ( ( f  =  F  /\  o  =  .<_  )  -> 
(ordTop `  o )  =  (ordTop `  .<_  ) )
2423opeq2d 4226 . . . . . . 7  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  <. (TopSet `  ndx ) ,  (ordTop `  o ) >.  =  <. (TopSet `  ndx ) ,  (ordTop `  .<_  )
>. )
2521, 24preq12d 4119 . . . . . 6  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  { <. ( Base `  ndx ) ,  f >. , 
<. (TopSet `  ndx ) ,  (ordTop `  o ) >. }  =  { <. (
Base `  ndx ) ,  F >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  .<_  ) >. } )
2622opeq2d 4226 . . . . . . 7  |-  ( ( f  =  F  /\  o  =  .<_  )  ->  <. ( le `  ndx ) ,  o >.  = 
<. ( le `  ndx ) ,  .<_  >. )
27 id 22 . . . . . . . . . 10  |-  ( f  =  F  ->  f  =  F )
28 rabeq 3103 . . . . . . . . . . 11  |-  ( f  =  F  ->  { y  e.  f  |  ( y  i^i  x )  =  (/) }  =  {
y  e.  F  | 
( y  i^i  x
)  =  (/) } )
2928unieqd 4261 . . . . . . . . . 10  |-  ( f  =  F  ->  U. {
y  e.  f  |  ( y  i^i  x
)  =  (/) }  =  U. { y  e.  F  |  ( y  i^i  x )  =  (/) } )
3027, 29mpteq12dv 4535 . . . . . . . . 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 4226 . . . . . . 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 4119 . . . . . 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 3655 . . . . 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 3458 . . . 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 15908 . . . 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 4698 . . . . 5  |-  { <. (
Base `  ndx ) ,  F >. ,  <. (TopSet ` 
ndx ) ,  (ordTop `  .<_  ) >. }  e.  _V
38 prex 4698 . . . . 5  |-  { <. ( le `  ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |-> 
U. { y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. }  e.  _V
3937, 38unex 6597 . . . 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 5956 . . 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 2510 . 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 1395    e. wcel 1819   {crab 2811   _Vcvv 3109   [_csb 3430    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   {cpr 4034   <.cop 4038   U.cuni 4251   {copab 4514    |-> cmpt 4515    X. cxp 5006   ` cfv 5594   ndxcnx 14640   Basecbs 14643  TopSetcts 14717   lecple 14718   occoc 14719  ordTopcordt 14915  toInccipo 15907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ipo 15908
This theorem is referenced by:  ipobas  15911  ipolerval  15912  ipotset  15913
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