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Theorem psssdm 17935
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 17934 . 2 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋𝐴))
3 sseqin2 4104 . . 3 (𝐴𝑋 ↔ (𝑋𝐴) = 𝐴)
43biimpi 219 . 2 (𝐴𝑋 → (𝑋𝐴) = 𝐴)
52, 4sylan9eq 2793 1 ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2113  cin 3840  wss 3841   × cxp 5517  dom cdm 5519  PosetRelcps 17917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ps 17919
This theorem is referenced by:  ordtrest2lem  21947  ordtrest2  21948  icopnfhmeo  23688  iccpnfhmeo  23690  xrhmeo  23691  xrge0iifhmeo  31450
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