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Theorem psssdm 17437
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 ∈ PosetRel ∧ 𝐴𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)

Proof of Theorem psssdm
StepHypRef Expression
1 psssdm.1 . . 3 𝑋 = dom 𝑅
21psssdm2 17436 . 2 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋𝐴))
3 sseqin2 3960 . . 3 (𝐴𝑋 ↔ (𝑋𝐴) = 𝐴)
43biimpi 206 . 2 (𝐴𝑋 → (𝑋𝐴) = 𝐴)
52, 4sylan9eq 2814 1 ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  cin 3714  wss 3715   × cxp 5264  dom cdm 5266  PosetRelcps 17419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ps 17421
This theorem is referenced by:  ordtrest2lem  21229  ordtrest2  21230  icopnfhmeo  22963  iccpnfhmeo  22965  xrhmeo  22966  xrge0iifhmeo  30312
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