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Theorem psssdm2 14324
Description: Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.)
Hypothesis
Ref Expression
psssdm.1  |-  X  =  dom  R
Assertion
Ref Expression
psssdm2  |-  ( R  e.  PosetRel  ->  dom  ( R  i^i  ( A  X.  A
) )  =  ( X  i^i  A ) )

Proof of Theorem psssdm2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dmin 4886 . . . 4  |-  dom  ( R  i^i  ( A  X.  A ) )  C_  ( dom  R  i^i  dom  ( A  X.  A
) )
2 psssdm.1 . . . . . 6  |-  X  =  dom  R
32eqcomi 2287 . . . . 5  |-  dom  R  =  X
4 dmxpid 4898 . . . . 5  |-  dom  ( A  X.  A )  =  A
53, 4ineq12i 3368 . . . 4  |-  ( dom 
R  i^i  dom  ( A  X.  A ) )  =  ( X  i^i  A )
61, 5sseqtri 3210 . . 3  |-  dom  ( R  i^i  ( A  X.  A ) )  C_  ( X  i^i  A )
76a1i 10 . 2  |-  ( R  e.  PosetRel  ->  dom  ( R  i^i  ( A  X.  A
) )  C_  ( X  i^i  A ) )
8 inss2 3390 . . . . . . 7  |-  ( X  i^i  A )  C_  A
9 simpr 447 . . . . . . 7  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x  e.  ( X  i^i  A
) )
108, 9sseldi 3178 . . . . . 6  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x  e.  A )
11 inss1 3389 . . . . . . . 8  |-  ( X  i^i  A )  C_  X
1211sseli 3176 . . . . . . 7  |-  ( x  e.  ( X  i^i  A )  ->  x  e.  X )
132psref 14317 . . . . . . 7  |-  ( ( R  e.  PosetRel  /\  x  e.  X )  ->  x R x )
1412, 13sylan2 460 . . . . . 6  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x R x )
15 brinxp2 4751 . . . . . 6  |-  ( x ( R  i^i  ( A  X.  A ) ) x  <->  ( x  e.  A  /\  x  e.  A  /\  x R x ) )
1610, 10, 14, 15syl3anbrc 1136 . . . . 5  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x
( R  i^i  ( A  X.  A ) ) x )
17 vex 2791 . . . . . 6  |-  x  e. 
_V
1817, 17breldm 4883 . . . . 5  |-  ( x ( R  i^i  ( A  X.  A ) ) x  ->  x  e.  dom  ( R  i^i  ( A  X.  A ) ) )
1916, 18syl 15 . . . 4  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x  e.  dom  ( R  i^i  ( A  X.  A
) ) )
2019ex 423 . . 3  |-  ( R  e.  PosetRel  ->  ( x  e.  ( X  i^i  A
)  ->  x  e.  dom  ( R  i^i  ( A  X.  A ) ) ) )
2120ssrdv 3185 . 2  |-  ( R  e.  PosetRel  ->  ( X  i^i  A )  C_  dom  ( R  i^i  ( A  X.  A ) ) )
227, 21eqssd 3196 1  |-  ( R  e.  PosetRel  ->  dom  ( R  i^i  ( A  X.  A
) )  =  ( X  i^i  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   class class class wbr 4023    X. cxp 4687   dom cdm 4689   PosetRelcps 14301
This theorem is referenced by:  psssdm  14325  ordtrest  16932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ps 14306
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