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Theorem psssdm2 14567
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 5010 . . . 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 2384 . . . . 5  |-  dom  R  =  X
4 dmxpid 5022 . . . . 5  |-  dom  ( A  X.  A )  =  A
53, 4ineq12i 3476 . . . 4  |-  ( dom 
R  i^i  dom  ( A  X.  A ) )  =  ( X  i^i  A )
61, 5sseqtri 3316 . . 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 3498 . . . . . . 7  |-  ( X  i^i  A )  C_  A
9 simpr 448 . . . . . . 7  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x  e.  ( X  i^i  A
) )
108, 9sseldi 3282 . . . . . 6  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x  e.  A )
11 inss1 3497 . . . . . . . 8  |-  ( X  i^i  A )  C_  X
1211sseli 3280 . . . . . . 7  |-  ( x  e.  ( X  i^i  A )  ->  x  e.  X )
132psref 14560 . . . . . . 7  |-  ( ( R  e.  PosetRel  /\  x  e.  X )  ->  x R x )
1412, 13sylan2 461 . . . . . 6  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x R x )
15 brinxp2 4872 . . . . . 6  |-  ( x ( R  i^i  ( A  X.  A ) ) x  <->  ( x  e.  A  /\  x  e.  A  /\  x R x ) )
1610, 10, 14, 15syl3anbrc 1138 . . . . 5  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x
( R  i^i  ( A  X.  A ) ) x )
17 vex 2895 . . . . . 6  |-  x  e. 
_V
1817, 17breldm 5007 . . . . 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 424 . . 3  |-  ( R  e.  PosetRel  ->  ( x  e.  ( X  i^i  A
)  ->  x  e.  dom  ( R  i^i  ( A  X.  A ) ) ) )
2120ssrdv 3290 . 2  |-  ( R  e.  PosetRel  ->  ( X  i^i  A )  C_  dom  ( R  i^i  ( A  X.  A ) ) )
227, 21eqssd 3301 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 359    = wceq 1649    e. wcel 1717    i^i cin 3255    C_ wss 3256   class class class wbr 4146    X. cxp 4809   dom cdm 4811   PosetRelcps 14544
This theorem is referenced by:  psssdm  14568  ordtrest  17181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ps 14549
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