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Theorem glbdm 15470
Description: Domain of the greatest lower bound function of a poset. (Contributed by NM, 6-Sep-2018.)
Hypotheses
Ref Expression
glbfval.b  |-  B  =  ( Base `  K
)
glbfval.l  |-  .<_  =  ( le `  K )
glbfval.g  |-  G  =  ( glb `  K
)
glbfval.p  |-  ( ps  <->  ( A. y  e.  s  x  .<_  y  /\  A. z  e.  B  ( A. y  e.  s  z  .<_  y  ->  z 
.<_  x ) ) )
glbfval.k  |-  ( ph  ->  K  e.  V )
Assertion
Ref Expression
glbdm  |-  ( ph  ->  dom  G  =  {
s  e.  ~P B  |  E! x  e.  B  ps } )
Distinct variable groups:    x, s,
z, B    y, s, K, x, z
Allowed substitution hints:    ph( x, y, z, s)    ps( x, y, z, s)    B( y)    G( x, y, z, s)    .<_ ( x, y, z, s)    V( x, y, z, s)

Proof of Theorem glbdm
StepHypRef Expression
1 glbfval.b . . . 4  |-  B  =  ( Base `  K
)
2 glbfval.l . . . 4  |-  .<_  =  ( le `  K )
3 glbfval.g . . . 4  |-  G  =  ( glb `  K
)
4 glbfval.p . . . 4  |-  ( ps  <->  ( A. y  e.  s  x  .<_  y  /\  A. z  e.  B  ( A. y  e.  s  z  .<_  y  ->  z 
.<_  x ) ) )
5 glbfval.k . . . 4  |-  ( ph  ->  K  e.  V )
61, 2, 3, 4, 5glbfval 15469 . . 3  |-  ( ph  ->  G  =  ( ( s  e.  ~P B  |->  ( iota_ x  e.  B  ps ) )  |`  { s  |  E! x  e.  B  ps } ) )
76dmeqd 5198 . 2  |-  ( ph  ->  dom  G  =  dom  ( ( s  e. 
~P B  |->  ( iota_ x  e.  B  ps )
)  |`  { s  |  E! x  e.  B  ps } ) )
8 riotaex 6242 . . . . 5  |-  ( iota_ x  e.  B  ps )  e.  _V
9 eqid 2462 . . . . 5  |-  ( s  e.  ~P B  |->  (
iota_ x  e.  B  ps ) )  =  ( s  e.  ~P B  |->  ( iota_ x  e.  B  ps ) )
108, 9dmmpti 5703 . . . 4  |-  dom  (
s  e.  ~P B  |->  ( iota_ x  e.  B  ps ) )  =  ~P B
1110ineq2i 3692 . . 3  |-  ( { s  |  E! x  e.  B  ps }  i^i  dom  ( s  e.  ~P B  |->  ( iota_ x  e.  B  ps ) ) )  =  ( { s  |  E! x  e.  B  ps }  i^i  ~P B )
12 dmres 5287 . . 3  |-  dom  (
( s  e.  ~P B  |->  ( iota_ x  e.  B  ps ) )  |`  { s  |  E! x  e.  B  ps } )  =  ( { s  |  E! x  e.  B  ps }  i^i  dom  ( s  e.  ~P B  |->  ( iota_ x  e.  B  ps )
) )
13 dfrab2 3769 . . 3  |-  { s  e.  ~P B  |  E! x  e.  B  ps }  =  ( { s  |  E! x  e.  B  ps }  i^i  ~P B )
1411, 12, 133eqtr4i 2501 . 2  |-  dom  (
( s  e.  ~P B  |->  ( iota_ x  e.  B  ps ) )  |`  { s  |  E! x  e.  B  ps } )  =  {
s  e.  ~P B  |  E! x  e.  B  ps }
157, 14syl6eq 2519 1  |-  ( ph  ->  dom  G  =  {
s  e.  ~P B  |  E! x  e.  B  ps } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   {cab 2447   A.wral 2809   E!wreu 2811   {crab 2813    i^i cin 3470   ~Pcpw 4005   class class class wbr 4442    |-> cmpt 4500   dom cdm 4994    |` cres 4996   ` cfv 5581   iota_crio 6237   Basecbs 14481   lecple 14553   glbcglb 15421
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-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-glb 15453
This theorem is referenced by:  glbeldm  15472  xrsclat  27318
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