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Theorem lubid 15264
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 3537 . . . 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 15258 . 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 15262 . . 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 6179 . 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 2492 1  |-  ( ph  ->  ( U `  {
y  e.  B  | 
y  .<_  X } )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   {crab 2799    C_ wss 3428   class class class wbr 4392   ` cfv 5518   iota_crio 6152   Basecbs 14278   lecple 14349   Posetcpo 15214   lubclub 15216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-poset 15220  df-lub 15248
This theorem is referenced by:  atlatmstc  33272
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