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Theorem lubid 15466
Description: The LUB of elements less than or equal to a fixed value equals that value. (Contributed by NM, 19-Oct-2011.) (Revised by NM, 7-Sep-2018.)
Hypotheses
Ref Expression
lubid.b  |-  B  =  ( Base `  K
)
lubid.l  |-  .<_  =  ( le `  K )
lubid.u  |-  U  =  ( lub `  K
)
lubid.k  |-  ( ph  ->  K  e.  Poset )
lubid.x  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
lubid  |-  ( ph  ->  ( U `  {
y  e.  B  | 
y  .<_  X } )  =  X )
Distinct variable groups:    y,  .<_    y, B    y, X
Allowed substitution hints:    ph( y)    U( y)    K( y)

Proof of Theorem lubid
Dummy variables  x  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lubid.b . . 3  |-  B  =  ( Base `  K
)
2 lubid.l . . 3  |-  .<_  =  ( le `  K )
3 lubid.u . . 3  |-  U  =  ( lub `  K
)
4 biid 236 . . 3  |-  ( ( A. z  e.  {
y  e.  B  | 
y  .<_  X } z 
.<_  x  /\  A. w  e.  B  ( A. z  e.  { y  e.  B  |  y  .<_  X } z  .<_  w  ->  x  .<_  w ) )  <->  ( A. z  e.  { y  e.  B  |  y  .<_  X }
z  .<_  x  /\  A. w  e.  B  ( A. z  e.  { y  e.  B  |  y 
.<_  X } z  .<_  w  ->  x  .<_  w ) ) )
5 lubid.k . . 3  |-  ( ph  ->  K  e.  Poset )
6 ssrab2 3578 . . . 4  |-  { y  e.  B  |  y 
.<_  X }  C_  B
76a1i 11 . . 3  |-  ( ph  ->  { y  e.  B  |  y  .<_  X }  C_  B )
81, 2, 3, 4, 5, 7lubval 15460 . 2  |-  ( ph  ->  ( U `  {
y  e.  B  | 
y  .<_  X } )  =  ( iota_ x  e.  B  ( A. z  e.  { y  e.  B  |  y  .<_  X }
z  .<_  x  /\  A. w  e.  B  ( A. z  e.  { y  e.  B  |  y 
.<_  X } z  .<_  w  ->  x  .<_  w ) ) ) )
9 lubid.x . . 3  |-  ( ph  ->  X  e.  B )
101, 2, 3, 5, 9lublecllem 15464 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
( A. z  e. 
{ y  e.  B  |  y  .<_  X }
z  .<_  x  /\  A. w  e.  B  ( A. z  e.  { y  e.  B  |  y 
.<_  X } z  .<_  w  ->  x  .<_  w ) )  <->  x  =  X
) )
119, 10riota5 6262 . 2  |-  ( ph  ->  ( iota_ x  e.  B  ( A. z  e.  {
y  e.  B  | 
y  .<_  X } z 
.<_  x  /\  A. w  e.  B  ( A. z  e.  { y  e.  B  |  y  .<_  X } z  .<_  w  ->  x  .<_  w ) ) )  =  X )
128, 11eqtrd 2501 1  |-  ( ph  ->  ( U `  {
y  e.  B  | 
y  .<_  X } )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811    C_ wss 3469   class class class wbr 4440   ` cfv 5579   iota_crio 6235   Basecbs 14479   lecple 14551   Posetcpo 15416   lubclub 15418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-poset 15422  df-lub 15450
This theorem is referenced by:  atlatmstc  33991
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