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Theorem lublecl 15476
Description: The set of all elements less than a given element has an LUB. (Contributed by NM, 8-Sep-2018.)
Hypotheses
Ref Expression
lublecl.b  |-  B  =  ( Base `  K
)
lublecl.l  |-  .<_  =  ( le `  K )
lublecl.u  |-  U  =  ( lub `  K
)
lublecl.k  |-  ( ph  ->  K  e.  Poset )
lublecl.x  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
lublecl  |-  ( ph  ->  { y  e.  B  |  y  .<_  X }  e.  dom  U )
Distinct variable groups:    y,  .<_    y, B    y, X
Allowed substitution hints:    ph( y)    U( y)    K( y)

Proof of Theorem lublecl
Dummy variables  x  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3585 . . 3  |-  { y  e.  B  |  y 
.<_  X }  C_  B
21a1i 11 . 2  |-  ( ph  ->  { y  e.  B  |  y  .<_  X }  C_  B )
3 lublecl.x . . 3  |-  ( ph  ->  X  e.  B )
4 lublecl.b . . . . 5  |-  B  =  ( Base `  K
)
5 lublecl.l . . . . 5  |-  .<_  =  ( le `  K )
6 lublecl.u . . . . 5  |-  U  =  ( lub `  K
)
7 lublecl.k . . . . 5  |-  ( ph  ->  K  e.  Poset )
84, 5, 6, 7, 3lublecllem 15475 . . . 4  |-  ( (
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
) )
98ralrimiva 2878 . . 3  |-  ( ph  ->  A. 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
) )
10 reu6i 3294 . . 3  |-  ( ( X  e.  B  /\  A. 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
) )  ->  E! 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 ) ) )
113, 9, 10syl2anc 661 . 2  |-  ( ph  ->  E! 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 ) ) )
12 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 ) ) )
134, 5, 6, 12, 7lubeldm 15468 . 2  |-  ( ph  ->  ( { y  e.  B  |  y  .<_  X }  e.  dom  U  <-> 
( { y  e.  B  |  y  .<_  X }  C_  B  /\  E! 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 ) ) ) ) )
142, 11, 13mpbir2and 920 1  |-  ( ph  ->  { y  e.  B  |  y  .<_  X }  e.  dom  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E!wreu 2816   {crab 2818    C_ wss 3476   class class class wbr 4447   dom cdm 4999   ` cfv 5588   Basecbs 14490   lecple 14562   Posetcpo 15427   lubclub 15429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-poset 15433  df-lub 15461
This theorem is referenced by: (None)
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