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Theorem psssdm2 18626
Description: Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.)
Hypothesis
Ref Expression
psssdm.1 𝑋 = dom 𝑅
Assertion
Ref Expression
psssdm2 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋𝐴))

Proof of Theorem psssdm2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dmin 5922 . . . 4 dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (dom 𝑅 ∩ dom (𝐴 × 𝐴))
2 psssdm.1 . . . . . 6 𝑋 = dom 𝑅
32eqcomi 2746 . . . . 5 dom 𝑅 = 𝑋
4 dmxpid 5941 . . . . 5 dom (𝐴 × 𝐴) = 𝐴
53, 4ineq12i 4218 . . . 4 (dom 𝑅 ∩ dom (𝐴 × 𝐴)) = (𝑋𝐴)
61, 5sseqtri 4032 . . 3 dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋𝐴)
76a1i 11 . 2 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋𝐴))
8 simpr 484 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥 ∈ (𝑋𝐴))
98elin2d 4205 . . . 4 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥𝐴)
10 elinel1 4201 . . . . 5 (𝑥 ∈ (𝑋𝐴) → 𝑥𝑋)
112psref 18619 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝑥𝑋) → 𝑥𝑅𝑥)
1210, 11sylan2 593 . . . 4 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥𝑅𝑥)
13 brinxp2 5763 . . . 4 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥𝐴𝑥𝐴) ∧ 𝑥𝑅𝑥))
149, 9, 12, 13syl21anbrc 1345 . . 3 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
15 vex 3484 . . . 4 𝑥 ∈ V
1615, 15breldm 5919 . . 3 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)))
1714, 16syl 17 . 2 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)))
187, 17eqelssd 4005 1 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cin 3950  wss 3951   class class class wbr 5143   × cxp 5683  dom cdm 5685  PosetRelcps 18609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ps 18611
This theorem is referenced by:  psssdm  18627  ordtrest  23210
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