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Theorem psssdm 14325
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 14324 . 2  |-  ( R  e.  PosetRel  ->  dom  ( R  i^i  ( A  X.  A
) )  =  ( X  i^i  A ) )
3 dfss1 3373 . . 3  |-  ( A 
C_  X  <->  ( X  i^i  A )  =  A )
43biimpi 186 . 2  |-  ( A 
C_  X  ->  ( X  i^i  A )  =  A )
52, 4sylan9eq 2335 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 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152    X. cxp 4687   dom cdm 4689   PosetRelcps 14301
This theorem is referenced by:  ordtrest2lem  16933  ordtrest2  16934  icopnfhmeo  18441  iccpnfhmeo  18443  xrhmeo  18444  xrge0iifhmeo  23318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ps 14306
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