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Theorem psssdm 16404
Description: Field of a subposet. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 9-Sep-2015.)
Hypothesis
Ref Expression
psssdm.1  |-  X  =  dom  R
Assertion
Ref Expression
psssdm  |-  ( ( R  e.  PosetRel  /\  A  C_  X )  ->  dom  ( R  i^i  ( A  X.  A ) )  =  A )

Proof of Theorem psssdm
StepHypRef Expression
1 psssdm.1 . . 3  |-  X  =  dom  R
21psssdm2 16403 . 2  |-  ( R  e.  PosetRel  ->  dom  ( R  i^i  ( A  X.  A
) )  =  ( X  i^i  A ) )
3 dfss1 3673 . . 3  |-  ( A 
C_  X  <->  ( X  i^i  A )  =  A )
43biimpi 197 . 2  |-  ( A 
C_  X  ->  ( X  i^i  A )  =  A )
52, 4sylan9eq 2490 1  |-  ( ( R  e.  PosetRel  /\  A  C_  X )  ->  dom  ( R  i^i  ( A  X.  A ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    i^i cin 3441    C_ wss 3442    X. cxp 4852   dom cdm 4854   PosetRelcps 16386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ps 16388
This theorem is referenced by:  ordtrest2lem  20141  ordtrest2  20142  icopnfhmeo  21858  iccpnfhmeo  21860  xrhmeo  21861  xrge0iifhmeo  28572
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