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Theorem prsdm 28720
Description: Domain of the relation of a preset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
Hypotheses
Ref Expression
ordtNEW.b  |-  B  =  ( Base `  K
)
ordtNEW.l  |-  .<_  =  ( ( le `  K
)  i^i  ( B  X.  B ) )
Assertion
Ref Expression
prsdm  |-  ( K  e.  Preset  ->  dom  .<_  =  B )

Proof of Theorem prsdm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtNEW.l . . . . 5  |-  .<_  =  ( ( le `  K
)  i^i  ( B  X.  B ) )
21dmeqi 5036 . . . 4  |-  dom  .<_  =  dom  ( ( le
`  K )  i^i  ( B  X.  B
) )
32eleq2i 2521 . . 3  |-  ( x  e.  dom  .<_  <->  x  e.  dom  ( ( le `  K )  i^i  ( B  X.  B ) ) )
4 ordtNEW.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
5 eqid 2451 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
64, 5prsref 16177 . . . . . . . . 9  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  x
( le `  K
) x )
7 df-br 4403 . . . . . . . . 9  |-  ( x ( le `  K
) x  <->  <. x ,  x >.  e.  ( le `  K ) )
86, 7sylib 200 . . . . . . . 8  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  <. x ,  x >.  e.  ( le `  K ) )
9 simpr 463 . . . . . . . . 9  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  x  e.  B )
10 opelxpi 4866 . . . . . . . . 9  |-  ( ( x  e.  B  /\  x  e.  B )  -> 
<. x ,  x >.  e.  ( B  X.  B
) )
119, 10sylancom 673 . . . . . . . 8  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  <. x ,  x >.  e.  ( B  X.  B ) )
128, 11elind 3618 . . . . . . 7  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  <. x ,  x >.  e.  (
( le `  K
)  i^i  ( B  X.  B ) ) )
13 vex 3048 . . . . . . . 8  |-  x  e. 
_V
14 opeq2 4167 . . . . . . . . 9  |-  ( y  =  x  ->  <. x ,  y >.  =  <. x ,  x >. )
1514eleq1d 2513 . . . . . . . 8  |-  ( y  =  x  ->  ( <. x ,  y >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) )  <->  <. x ,  x >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) ) ) )
1613, 15spcev 3141 . . . . . . 7  |-  ( <.
x ,  x >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) )  ->  E. y <. x ,  y >.  e.  ( ( le `  K
)  i^i  ( B  X.  B ) ) )
1712, 16syl 17 . . . . . 6  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  E. y <. x ,  y >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) ) )
1817ex 436 . . . . 5  |-  ( K  e.  Preset  ->  ( x  e.  B  ->  E. y <. x ,  y >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) ) ) )
19 inss2 3653 . . . . . . . 8  |-  ( ( le `  K )  i^i  ( B  X.  B ) )  C_  ( B  X.  B
)
2019sseli 3428 . . . . . . 7  |-  ( <.
x ,  y >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) )  ->  <. x ,  y
>.  e.  ( B  X.  B ) )
21 opelxp1 4867 . . . . . . 7  |-  ( <.
x ,  y >.  e.  ( B  X.  B
)  ->  x  e.  B )
2220, 21syl 17 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) )  ->  x  e.  B
)
2322exlimiv 1776 . . . . 5  |-  ( E. y <. x ,  y
>.  e.  ( ( le
`  K )  i^i  ( B  X.  B
) )  ->  x  e.  B )
2418, 23impbid1 207 . . . 4  |-  ( K  e.  Preset  ->  ( x  e.  B  <->  E. y <. x ,  y >.  e.  ( ( le `  K
)  i^i  ( B  X.  B ) ) ) )
2513eldm2 5033 . . . 4  |-  ( x  e.  dom  ( ( le `  K )  i^i  ( B  X.  B ) )  <->  E. y <. x ,  y >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) ) )
2624, 25syl6rbbr 268 . . 3  |-  ( K  e.  Preset  ->  ( x  e. 
dom  ( ( le
`  K )  i^i  ( B  X.  B
) )  <->  x  e.  B ) )
273, 26syl5bb 261 . 2  |-  ( K  e.  Preset  ->  ( x  e. 
dom  .<_ 
<->  x  e.  B ) )
2827eqrdv 2449 1  |-  ( K  e.  Preset  ->  dom  .<_  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887    i^i cin 3403   <.cop 3974   class class class wbr 4402    X. cxp 4832   dom cdm 4834   ` cfv 5582   Basecbs 15121   lecple 15197    Preset cpreset 16171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-xp 4840  df-dm 4844  df-iota 5546  df-fv 5590  df-preset 16173
This theorem is referenced by:  prsssdm  28723  ordtprsval  28724  ordtprsuni  28725  ordtrestNEW  28727  ordtconlem1  28730
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