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Theorem psssdm2 14639
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 5069 . . . 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 2439 . . . . 5  |-  dom  R  =  X
4 dmxpid 5081 . . . . 5  |-  dom  ( A  X.  A )  =  A
53, 4ineq12i 3532 . . . 4  |-  ( dom 
R  i^i  dom  ( A  X.  A ) )  =  ( X  i^i  A )
61, 5sseqtri 3372 . . 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 3554 . . . . . . 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 3338 . . . . . 6  |-  ( ( R  e.  PosetRel  /\  x  e.  ( X  i^i  A
) )  ->  x  e.  A )
11 inss1 3553 . . . . . . . 8  |-  ( X  i^i  A )  C_  X
1211sseli 3336 . . . . . . 7  |-  ( x  e.  ( X  i^i  A )  ->  x  e.  X )
132psref 14632 . . . . . . 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 4931 . . . . . 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 2951 . . . . . 6  |-  x  e. 
_V
1817, 17breldm 5066 . . . . 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 3346 . 2  |-  ( R  e.  PosetRel  ->  ( X  i^i  A )  C_  dom  ( R  i^i  ( A  X.  A ) ) )
227, 21eqssd 3357 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 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312   class class class wbr 4204    X. cxp 4868   dom cdm 4870   PosetRelcps 14616
This theorem is referenced by:  psssdm  14640  ordtrest  17258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ps 14621
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