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Theorem psssdm2 14340
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 4902 . . . 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 2300 . . . . 5  |-  dom  R  =  X
4 dmxpid 4914 . . . . 5  |-  dom  ( A  X.  A )  =  A
53, 4ineq12i 3381 . . . 4  |-  ( dom 
R  i^i  dom  ( A  X.  A ) )  =  ( X  i^i  A )
61, 5sseqtri 3223 . . 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 3403 . . . . . . 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 3191 . . . . . 6  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x  e.  A )
11 inss1 3402 . . . . . . . 8  |-  ( X  i^i  A )  C_  X
1211sseli 3189 . . . . . . 7  |-  ( x  e.  ( X  i^i  A )  ->  x  e.  X )
132psref 14333 . . . . . . 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 4767 . . . . . 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 2804 . . . . . 6  |-  x  e. 
_V
1817, 17breldm 4899 . . . . 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 3198 . 2  |-  ( R  e.  PosetRel  ->  ( X  i^i  A )  C_  dom  ( R  i^i  ( A  X.  A ) ) )
227, 21eqssd 3209 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 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   class class class wbr 4039    X. cxp 4703   dom cdm 4705   PosetRelcps 14317
This theorem is referenced by:  psssdm  14341  ordtrest  16948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ps 14322
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