MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  joineu Structured version   Unicode version

Theorem joineu 15767
Description: Uniqueness of join of elements in the domain. (Contributed by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
joinval2.b  |-  B  =  ( Base `  K
)
joinval2.l  |-  .<_  =  ( le `  K )
joinval2.j  |-  .\/  =  ( join `  K )
joinval2.k  |-  ( ph  ->  K  e.  V )
joinval2.x  |-  ( ph  ->  X  e.  B )
joinval2.y  |-  ( ph  ->  Y  e.  B )
joinlem.e  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  .\/  )
Assertion
Ref Expression
joineu  |-  ( ph  ->  E! x  e.  B  ( ( X  .<_  x  /\  Y  .<_  x )  /\  A. z  e.  B  ( ( X 
.<_  z  /\  Y  .<_  z )  ->  x  .<_  z ) ) )
Distinct variable groups:    x, z, B    x,  .\/ , z    x, K, z    x, X, z   
x, Y, z    ph, x
Allowed substitution hints:    ph( z)    .<_ ( x, z)    V( x, z)

Proof of Theorem joineu
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 joinlem.e . 2  |-  ( ph  -> 
<. X ,  Y >.  e. 
dom  .\/  )
2 eqid 2457 . . . 4  |-  ( lub `  K )  =  ( lub `  K )
3 joinval2.j . . . 4  |-  .\/  =  ( join `  K )
4 joinval2.k . . . 4  |-  ( ph  ->  K  e.  V )
5 joinval2.x . . . 4  |-  ( ph  ->  X  e.  B )
6 joinval2.y . . . 4  |-  ( ph  ->  Y  e.  B )
72, 3, 4, 5, 6joindef 15761 . . 3  |-  ( ph  ->  ( <. X ,  Y >.  e.  dom  .\/  <->  { X ,  Y }  e.  dom  ( lub `  K ) ) )
8 joinval2.b . . . . . 6  |-  B  =  ( Base `  K
)
9 joinval2.l . . . . . 6  |-  .<_  =  ( le `  K )
10 biid 236 . . . . . 6  |-  ( ( A. y  e.  { X ,  Y }
y  .<_  x  /\  A. z  e.  B  ( A. y  e.  { X ,  Y } y  .<_  z  ->  x  .<_  z ) )  <->  ( A. y  e.  { X ,  Y } y  .<_  x  /\  A. z  e.  B  ( A. y  e.  { X ,  Y }
y  .<_  z  ->  x  .<_  z ) ) )
114adantr 465 . . . . . 6  |-  ( (
ph  /\  { X ,  Y }  e.  dom  ( lub `  K ) )  ->  K  e.  V )
12 simpr 461 . . . . . 6  |-  ( (
ph  /\  { X ,  Y }  e.  dom  ( lub `  K ) )  ->  { X ,  Y }  e.  dom  ( lub `  K ) )
138, 9, 2, 10, 11, 12lubeu 15740 . . . . 5  |-  ( (
ph  /\  { X ,  Y }  e.  dom  ( lub `  K ) )  ->  E! x  e.  B  ( A. y  e.  { X ,  Y } y  .<_  x  /\  A. z  e.  B  ( A. y  e.  { X ,  Y } y  .<_  z  ->  x  .<_  z ) ) )
1413ex 434 . . . 4  |-  ( ph  ->  ( { X ,  Y }  e.  dom  ( lub `  K )  ->  E! x  e.  B  ( A. y  e.  { X ,  Y } y  .<_  x  /\  A. z  e.  B  ( A. y  e.  { X ,  Y }
y  .<_  z  ->  x  .<_  z ) ) ) )
158, 9, 3, 4, 5, 6joinval2lem 15765 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( ( A. y  e.  { X ,  Y } y  .<_  x  /\  A. z  e.  B  ( A. y  e.  { X ,  Y }
y  .<_  z  ->  x  .<_  z ) )  <->  ( ( X  .<_  x  /\  Y  .<_  x )  /\  A. z  e.  B  (
( X  .<_  z  /\  Y  .<_  z )  ->  x  .<_  z ) ) ) )
165, 6, 15syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( A. y  e.  { X ,  Y } y  .<_  x  /\  A. z  e.  B  ( A. y  e.  { X ,  Y }
y  .<_  z  ->  x  .<_  z ) )  <->  ( ( X  .<_  x  /\  Y  .<_  x )  /\  A. z  e.  B  (
( X  .<_  z  /\  Y  .<_  z )  ->  x  .<_  z ) ) ) )
1716reubidv 3042 . . . 4  |-  ( ph  ->  ( E! x  e.  B  ( A. y  e.  { X ,  Y } y  .<_  x  /\  A. z  e.  B  ( A. y  e.  { X ,  Y }
y  .<_  z  ->  x  .<_  z ) )  <->  E! x  e.  B  ( ( X  .<_  x  /\  Y  .<_  x )  /\  A. z  e.  B  (
( X  .<_  z  /\  Y  .<_  z )  ->  x  .<_  z ) ) ) )
1814, 17sylibd 214 . . 3  |-  ( ph  ->  ( { X ,  Y }  e.  dom  ( lub `  K )  ->  E! x  e.  B  ( ( X 
.<_  x  /\  Y  .<_  x )  /\  A. z  e.  B  ( ( X  .<_  z  /\  Y  .<_  z )  ->  x  .<_  z ) ) ) )
197, 18sylbid 215 . 2  |-  ( ph  ->  ( <. X ,  Y >.  e.  dom  .\/  ->  E! x  e.  B  ( ( X  .<_  x  /\  Y  .<_  x )  /\  A. z  e.  B  ( ( X  .<_  z  /\  Y  .<_  z )  ->  x  .<_  z ) ) ) )
201, 19mpd 15 1  |-  ( ph  ->  E! x  e.  B  ( ( X  .<_  x  /\  Y  .<_  x )  /\  A. z  e.  B  ( ( X 
.<_  z  /\  Y  .<_  z )  ->  x  .<_  z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E!wreu 2809   {cpr 4034   <.cop 4038   class class class wbr 4456   dom cdm 5008   ` cfv 5594   Basecbs 14644   lecple 14719   lubclub 15698   joincjn 15700
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-rep 4568  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-reu 2814  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-iun 4334  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-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-oprab 6300  df-lub 15731  df-join 15733
This theorem is referenced by:  joinlem  15768
  Copyright terms: Public domain W3C validator