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Theorem prsssdm 28134
Description: Domain of a subpreset relation. (Contributed by Thierry Arnoux, 12-Sep-2018.)
Hypotheses
Ref Expression
ordtNEW.b  |-  B  =  ( Base `  K
)
ordtNEW.l  |-  .<_  =  ( ( le `  K
)  i^i  ( B  X.  B ) )
Assertion
Ref Expression
prsssdm  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  dom  (  .<_  i^i  ( A  X.  A ) )  =  A )

Proof of Theorem prsssdm
StepHypRef Expression
1 ordtNEW.b . . . 4  |-  B  =  ( Base `  K
)
2 ordtNEW.l . . . 4  |-  .<_  =  ( ( le `  K
)  i^i  ( B  X.  B ) )
31, 2prsss 28133 . . 3  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  (  .<_  i^i  ( A  X.  A ) )  =  ( ( le `  K )  i^i  ( A  X.  A ) ) )
43dmeqd 5194 . 2  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  dom  (  .<_  i^i  ( A  X.  A ) )  =  dom  ( ( le
`  K )  i^i  ( A  X.  A
) ) )
51ressprs 27877 . . . 4  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  ( Ks  A )  e.  Preset  )
6 eqid 2454 . . . . 5  |-  ( Base `  ( Ks  A ) )  =  ( Base `  ( Ks  A ) )
7 eqid 2454 . . . . 5  |-  ( ( le `  ( Ks  A ) )  i^i  (
( Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) )  =  ( ( le `  ( Ks  A ) )  i^i  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )
86, 7prsdm 28131 . . . 4  |-  ( ( Ks  A )  e.  Preset  ->  dom  ( ( le `  ( Ks  A ) )  i^i  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )  =  ( Base `  ( Ks  A ) ) )
95, 8syl 16 . . 3  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  dom  ( ( le `  ( Ks  A ) )  i^i  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )  =  ( Base `  ( Ks  A ) ) )
10 eqid 2454 . . . . . . . . 9  |-  ( Ks  A )  =  ( Ks  A )
1110, 1ressbas2 14774 . . . . . . . 8  |-  ( A 
C_  B  ->  A  =  ( Base `  ( Ks  A ) ) )
12 fvex 5858 . . . . . . . 8  |-  ( Base `  ( Ks  A ) )  e. 
_V
1311, 12syl6eqel 2550 . . . . . . 7  |-  ( A 
C_  B  ->  A  e.  _V )
14 eqid 2454 . . . . . . . 8  |-  ( le
`  K )  =  ( le `  K
)
1510, 14ressle 14888 . . . . . . 7  |-  ( A  e.  _V  ->  ( le `  K )  =  ( le `  ( Ks  A ) ) )
1613, 15syl 16 . . . . . 6  |-  ( A 
C_  B  ->  ( le `  K )  =  ( le `  ( Ks  A ) ) )
1716adantl 464 . . . . 5  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  ( le `  K )  =  ( le `  ( Ks  A ) ) )
1811adantl 464 . . . . . 6  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  A  =  ( Base `  ( Ks  A ) ) )
1918sqxpeqd 5014 . . . . 5  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  ( A  X.  A )  =  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )
2017, 19ineq12d 3687 . . . 4  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  (
( le `  K
)  i^i  ( A  X.  A ) )  =  ( ( le `  ( Ks  A ) )  i^i  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) ) )
2120dmeqd 5194 . . 3  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  dom  ( ( le `  K )  i^i  ( A  X.  A ) )  =  dom  ( ( le `  ( Ks  A ) )  i^i  (
( Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) ) )
229, 21, 183eqtr4d 2505 . 2  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  dom  ( ( le `  K )  i^i  ( A  X.  A ) )  =  A )
234, 22eqtrd 2495 1  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  dom  (  .<_  i^i  ( A  X.  A ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    i^i cin 3460    C_ wss 3461    X. cxp 4986   dom cdm 4988   ` cfv 5570  (class class class)co 6270   Basecbs 14716   ↾s cress 14717   lecple 14791    Preset cpreset 15754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-ple 14804  df-preset 15756
This theorem is referenced by:  ordtrest2NEWlem  28139  ordtrest2NEW  28140
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