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Theorem psssdm 14607
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 14606 . 2  |-  ( R  e.  PosetRel  ->  dom  ( R  i^i  ( A  X.  A
) )  =  ( X  i^i  A ) )
3 dfss1 3509 . . 3  |-  ( A 
C_  X  <->  ( X  i^i  A )  =  A )
43biimpi 187 . 2  |-  ( A 
C_  X  ->  ( X  i^i  A )  =  A )
52, 4sylan9eq 2460 1  |-  ( ( R  e.  PosetRel  /\  A  C_  X )  ->  dom  ( R  i^i  ( A  X.  A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3283    C_ wss 3284    X. cxp 4839   dom cdm 4841   PosetRelcps 14583
This theorem is referenced by:  ordtrest2lem  17225  ordtrest2  17226  icopnfhmeo  18925  iccpnfhmeo  18927  xrhmeo  18928  xrge0iifhmeo  24279
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ps 14588
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