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Theorem psssdm2 17147
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 5297 . . . 4 dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (dom 𝑅 ∩ dom (𝐴 × 𝐴))
2 psssdm.1 . . . . . 6 𝑋 = dom 𝑅
32eqcomi 2630 . . . . 5 dom 𝑅 = 𝑋
4 dmxpid 5310 . . . . 5 dom (𝐴 × 𝐴) = 𝐴
53, 4ineq12i 3795 . . . 4 (dom 𝑅 ∩ dom (𝐴 × 𝐴)) = (𝑋𝐴)
61, 5sseqtri 3621 . . 3 dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋𝐴)
76a1i 11 . 2 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋𝐴))
8 inss2 3817 . . . . . . 7 (𝑋𝐴) ⊆ 𝐴
9 simpr 477 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥 ∈ (𝑋𝐴))
108, 9sseldi 3585 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥𝐴)
11 inss1 3816 . . . . . . . 8 (𝑋𝐴) ⊆ 𝑋
1211sseli 3583 . . . . . . 7 (𝑥 ∈ (𝑋𝐴) → 𝑥𝑋)
132psref 17140 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝑥𝑋) → 𝑥𝑅𝑥)
1412, 13sylan2 491 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥𝑅𝑥)
15 brinxp2 5146 . . . . . 6 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥𝐴𝑥𝐴𝑥𝑅𝑥))
1610, 10, 14, 15syl3anbrc 1244 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
17 vex 3192 . . . . . 6 𝑥 ∈ V
1817, 17breldm 5294 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)))
1916, 18syl 17 . . . 4 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)))
2019ex 450 . . 3 (𝑅 ∈ PosetRel → (𝑥 ∈ (𝑋𝐴) → 𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))))
2120ssrdv 3593 . 2 (𝑅 ∈ PosetRel → (𝑋𝐴) ⊆ dom (𝑅 ∩ (𝐴 × 𝐴)))
227, 21eqssd 3604 1 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cin 3558  wss 3559   class class class wbr 4618   × cxp 5077  dom cdm 5079  PosetRelcps 17130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ps 17132
This theorem is referenced by:  psssdm  17148  ordtrest  20929
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