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Theorem psssdm2 15718
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 5216 . . . 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 2480 . . . . 5  |-  dom  R  =  X
4 dmxpid 5228 . . . . 5  |-  dom  ( A  X.  A )  =  A
53, 4ineq12i 3703 . . . 4  |-  ( dom 
R  i^i  dom  ( A  X.  A ) )  =  ( X  i^i  A )
61, 5sseqtri 3541 . . 3  |-  dom  ( R  i^i  ( A  X.  A ) )  C_  ( X  i^i  A )
76a1i 11 . 2  |-  ( R  e.  PosetRel  ->  dom  ( R  i^i  ( A  X.  A
) )  C_  ( X  i^i  A ) )
8 inss2 3724 . . . . . . 7  |-  ( X  i^i  A )  C_  A
9 simpr 461 . . . . . . 7  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x  e.  ( X  i^i  A
) )
108, 9sseldi 3507 . . . . . 6  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x  e.  A )
11 inss1 3723 . . . . . . . 8  |-  ( X  i^i  A )  C_  X
1211sseli 3505 . . . . . . 7  |-  ( x  e.  ( X  i^i  A )  ->  x  e.  X )
132psref 15711 . . . . . . 7  |-  ( ( R  e.  PosetRel  /\  x  e.  X )  ->  x R x )
1412, 13sylan2 474 . . . . . 6  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x R x )
15 brinxp2 5067 . . . . . 6  |-  ( x ( R  i^i  ( A  X.  A ) ) x  <->  ( x  e.  A  /\  x  e.  A  /\  x R x ) )
1610, 10, 14, 15syl3anbrc 1180 . . . . 5  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x
( R  i^i  ( A  X.  A ) ) x )
17 vex 3121 . . . . . 6  |-  x  e. 
_V
1817, 17breldm 5213 . . . . 5  |-  ( x ( R  i^i  ( A  X.  A ) ) x  ->  x  e.  dom  ( R  i^i  ( A  X.  A ) ) )
1916, 18syl 16 . . . 4  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x  e.  dom  ( R  i^i  ( A  X.  A
) ) )
2019ex 434 . . 3  |-  ( R  e.  PosetRel  ->  ( x  e.  ( X  i^i  A
)  ->  x  e.  dom  ( R  i^i  ( A  X.  A ) ) ) )
2120ssrdv 3515 . 2  |-  ( R  e.  PosetRel  ->  ( X  i^i  A )  C_  dom  ( R  i^i  ( A  X.  A ) ) )
227, 21eqssd 3526 1  |-  ( R  e.  PosetRel  ->  dom  ( R  i^i  ( A  X.  A
) )  =  ( X  i^i  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481   class class class wbr 4453    X. cxp 5003   dom cdm 5005   PosetRelcps 15701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ps 15703
This theorem is referenced by:  psssdm  15719  ordtrest  19569
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