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Theorem prsdm 28144
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 5214 . . . 4  |-  dom  .<_  =  dom  ( ( le
`  K )  i^i  ( B  X.  B
) )
32eleq2i 2535 . . 3  |-  ( x  e.  dom  .<_  <->  x  e.  dom  ( ( le `  K )  i^i  ( B  X.  B ) ) )
4 ordtNEW.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
5 eqid 2457 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
64, 5prsref 15779 . . . . . . . . 9  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  x
( le `  K
) x )
7 df-br 4457 . . . . . . . . 9  |-  ( x ( le `  K
) x  <->  <. x ,  x >.  e.  ( le `  K ) )
86, 7sylib 196 . . . . . . . 8  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  <. x ,  x >.  e.  ( le `  K ) )
9 simpr 461 . . . . . . . . 9  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  x  e.  B )
10 opelxpi 5040 . . . . . . . . 9  |-  ( ( x  e.  B  /\  x  e.  B )  -> 
<. x ,  x >.  e.  ( B  X.  B
) )
119, 10sylancom 667 . . . . . . . 8  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  <. x ,  x >.  e.  ( B  X.  B ) )
128, 11elind 3684 . . . . . . 7  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  <. x ,  x >.  e.  (
( le `  K
)  i^i  ( B  X.  B ) ) )
13 vex 3112 . . . . . . . 8  |-  x  e. 
_V
14 opeq2 4220 . . . . . . . . 9  |-  ( y  =  x  ->  <. x ,  y >.  =  <. x ,  x >. )
1514eleq1d 2526 . . . . . . . 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 3201 . . . . . . 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 16 . . . . . 6  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  E. y <. x ,  y >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) ) )
1817ex 434 . . . . 5  |-  ( K  e.  Preset  ->  ( x  e.  B  ->  E. y <. x ,  y >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) ) ) )
19 inss2 3715 . . . . . . . 8  |-  ( ( le `  K )  i^i  ( B  X.  B ) )  C_  ( B  X.  B
)
2019sseli 3495 . . . . . . 7  |-  ( <.
x ,  y >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) )  ->  <. x ,  y
>.  e.  ( B  X.  B ) )
21 opelxp1 5041 . . . . . . 7  |-  ( <.
x ,  y >.  e.  ( B  X.  B
)  ->  x  e.  B )
2220, 21syl 16 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) )  ->  x  e.  B
)
2322exlimiv 1723 . . . . 5  |-  ( E. y <. x ,  y
>.  e.  ( ( le
`  K )  i^i  ( B  X.  B
) )  ->  x  e.  B )
2418, 23impbid1 203 . . . 4  |-  ( K  e.  Preset  ->  ( x  e.  B  <->  E. y <. x ,  y >.  e.  ( ( le `  K
)  i^i  ( B  X.  B ) ) ) )
2513eldm2 5211 . . . 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 264 . . 3  |-  ( K  e.  Preset  ->  ( x  e. 
dom  ( ( le
`  K )  i^i  ( B  X.  B
) )  <->  x  e.  B ) )
273, 26syl5bb 257 . 2  |-  ( K  e.  Preset  ->  ( x  e. 
dom  .<_ 
<->  x  e.  B ) )
2827eqrdv 2454 1  |-  ( K  e.  Preset  ->  dom  .<_  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819    i^i cin 3470   <.cop 4038   class class class wbr 4456    X. cxp 5006   dom cdm 5008   ` cfv 5594   Basecbs 14735   lecple 14810    Preset cpreset 15773
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-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-dm 5018  df-iota 5557  df-fv 5602  df-preset 15775
This theorem is referenced by:  prsssdm  28147  ordtprsval  28148  ordtprsuni  28149  ordtrestNEW  28151  ordtconlem1  28154
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