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Theorem meetdmss 15629
Description: Subset property of domain of meet. (Contributed by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
meetdmss.b  |-  B  =  ( Base `  K
)
meetdmss.j  |-  ./\  =  ( meet `  K )
meetdmss.k  |-  ( ph  ->  K  e.  V )
Assertion
Ref Expression
meetdmss  |-  ( ph  ->  dom  ./\  C_  ( B  X.  B ) )

Proof of Theorem meetdmss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 5119 . . 3  |-  Rel  { <. x ,  y >.  |  { x ,  y }  e.  dom  ( glb `  K ) }
2 meetdmss.k . . . . 5  |-  ( ph  ->  K  e.  V )
3 eqid 2443 . . . . . 6  |-  ( glb `  K )  =  ( glb `  K )
4 meetdmss.j . . . . . 6  |-  ./\  =  ( meet `  K )
53, 4meetdm 15625 . . . . 5  |-  ( K  e.  V  ->  dom  ./\  =  { <. x ,  y >.  |  {
x ,  y }  e.  dom  ( glb `  K ) } )
62, 5syl 16 . . . 4  |-  ( ph  ->  dom  ./\  =  { <. x ,  y >.  |  { x ,  y }  e.  dom  ( glb `  K ) } )
76releqd 5077 . . 3  |-  ( ph  ->  ( Rel  dom  ./\  <->  Rel  { <. x ,  y >.  |  {
x ,  y }  e.  dom  ( glb `  K ) } ) )
81, 7mpbiri 233 . 2  |-  ( ph  ->  Rel  dom  ./\  )
9 vex 3098 . . . . 5  |-  x  e. 
_V
109a1i 11 . . . 4  |-  ( ph  ->  x  e.  _V )
11 vex 3098 . . . . 5  |-  y  e. 
_V
1211a1i 11 . . . 4  |-  ( ph  ->  y  e.  _V )
133, 4, 2, 10, 12meetdef 15626 . . 3  |-  ( ph  ->  ( <. x ,  y
>.  e.  dom  ./\  <->  { x ,  y }  e.  dom  ( glb `  K
) ) )
14 meetdmss.b . . . . . 6  |-  B  =  ( Base `  K
)
15 eqid 2443 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
162adantr 465 . . . . . 6  |-  ( (
ph  /\  { x ,  y }  e.  dom  ( glb `  K
) )  ->  K  e.  V )
17 simpr 461 . . . . . 6  |-  ( (
ph  /\  { x ,  y }  e.  dom  ( glb `  K
) )  ->  { x ,  y }  e.  dom  ( glb `  K
) )
1814, 15, 3, 16, 17glbelss 15603 . . . . 5  |-  ( (
ph  /\  { x ,  y }  e.  dom  ( glb `  K
) )  ->  { x ,  y }  C_  B )
1918ex 434 . . . 4  |-  ( ph  ->  ( { x ,  y }  e.  dom  ( glb `  K )  ->  { x ,  y }  C_  B
) )
209, 11prss 4169 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  B )  <->  { x ,  y } 
C_  B )
21 opelxpi 5021 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  B )  -> 
<. x ,  y >.  e.  ( B  X.  B
) )
2220, 21sylbir 213 . . . 4  |-  ( { x ,  y } 
C_  B  ->  <. x ,  y >.  e.  ( B  X.  B ) )
2319, 22syl6 33 . . 3  |-  ( ph  ->  ( { x ,  y }  e.  dom  ( glb `  K )  ->  <. x ,  y
>.  e.  ( B  X.  B ) ) )
2413, 23sylbid 215 . 2  |-  ( ph  ->  ( <. x ,  y
>.  e.  dom  ./\  ->  <.
x ,  y >.  e.  ( B  X.  B
) ) )
258, 24relssdv 5085 1  |-  ( ph  ->  dom  ./\  C_  ( B  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095    C_ wss 3461   {cpr 4016   <.cop 4020   {copab 4494    X. cxp 4987   dom cdm 4989   Rel wrel 4994   ` cfv 5578   Basecbs 14613   lecple 14685   glbcglb 15550   meetcmee 15552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-oprab 6285  df-glb 15583  df-meet 15585
This theorem is referenced by:  clatl  15724
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