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Theorem prsdm 28794
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 5041 . . . 4  |-  dom  .<_  =  dom  ( ( le
`  K )  i^i  ( B  X.  B
) )
32eleq2i 2541 . . 3  |-  ( x  e.  dom  .<_  <->  x  e.  dom  ( ( le `  K )  i^i  ( B  X.  B ) ) )
4 ordtNEW.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
5 eqid 2471 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
64, 5prsref 16255 . . . . . . . . 9  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  x
( le `  K
) x )
7 df-br 4396 . . . . . . . . 9  |-  ( x ( le `  K
) x  <->  <. x ,  x >.  e.  ( le `  K ) )
86, 7sylib 201 . . . . . . . 8  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  <. x ,  x >.  e.  ( le `  K ) )
9 simpr 468 . . . . . . . . 9  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  x  e.  B )
10 opelxpi 4871 . . . . . . . . 9  |-  ( ( x  e.  B  /\  x  e.  B )  -> 
<. x ,  x >.  e.  ( B  X.  B
) )
119, 10sylancom 680 . . . . . . . 8  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  <. x ,  x >.  e.  ( B  X.  B ) )
128, 11elind 3609 . . . . . . 7  |-  ( ( K  e.  Preset  /\  x  e.  B )  ->  <. x ,  x >.  e.  (
( le `  K
)  i^i  ( B  X.  B ) ) )
13 vex 3034 . . . . . . . 8  |-  x  e. 
_V
14 opeq2 4159 . . . . . . . . 9  |-  ( y  =  x  ->  <. x ,  y >.  =  <. x ,  x >. )
1514eleq1d 2533 . . . . . . . 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 3127 . . . . . . 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 441 . . . . 5  |-  ( K  e.  Preset  ->  ( x  e.  B  ->  E. y <. x ,  y >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) ) ) )
19 inss2 3644 . . . . . . . 8  |-  ( ( le `  K )  i^i  ( B  X.  B ) )  C_  ( B  X.  B
)
2019sseli 3414 . . . . . . 7  |-  ( <.
x ,  y >.  e.  ( ( le `  K )  i^i  ( B  X.  B ) )  ->  <. x ,  y
>.  e.  ( B  X.  B ) )
21 opelxp1 4872 . . . . . . 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 1784 . . . . 5  |-  ( E. y <. x ,  y
>.  e.  ( ( le
`  K )  i^i  ( B  X.  B
) )  ->  x  e.  B )
2418, 23impbid1 208 . . . 4  |-  ( K  e.  Preset  ->  ( x  e.  B  <->  E. y <. x ,  y >.  e.  ( ( le `  K
)  i^i  ( B  X.  B ) ) ) )
2513eldm2 5038 . . . 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 272 . . 3  |-  ( K  e.  Preset  ->  ( x  e. 
dom  ( ( le
`  K )  i^i  ( B  X.  B
) )  <->  x  e.  B ) )
273, 26syl5bb 265 . 2  |-  ( K  e.  Preset  ->  ( x  e. 
dom  .<_ 
<->  x  e.  B ) )
2827eqrdv 2469 1  |-  ( K  e.  Preset  ->  dom  .<_  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904    i^i cin 3389   <.cop 3965   class class class wbr 4395    X. cxp 4837   dom cdm 4839   ` cfv 5589   Basecbs 15199   lecple 15275    Preset cpreset 16249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-xp 4845  df-dm 4849  df-iota 5553  df-fv 5597  df-preset 16251
This theorem is referenced by:  prsssdm  28797  ordtprsval  28798  ordtprsuni  28799  ordtrestNEW  28801  ordtconlem1  28804
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