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Theorem psssdm2 17416
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 5468 . . . 4 dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (dom 𝑅 ∩ dom (𝐴 × 𝐴))
2 psssdm.1 . . . . . 6 𝑋 = dom 𝑅
32eqcomi 2780 . . . . 5 dom 𝑅 = 𝑋
4 dmxpid 5481 . . . . 5 dom (𝐴 × 𝐴) = 𝐴
53, 4ineq12i 3963 . . . 4 (dom 𝑅 ∩ dom (𝐴 × 𝐴)) = (𝑋𝐴)
61, 5sseqtri 3786 . . 3 dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋𝐴)
76a1i 11 . 2 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋𝐴))
8 inss2 3982 . . . . . . 7 (𝑋𝐴) ⊆ 𝐴
9 simpr 471 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥 ∈ (𝑋𝐴))
108, 9sseldi 3750 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥𝐴)
11 inss1 3981 . . . . . . . 8 (𝑋𝐴) ⊆ 𝑋
1211sseli 3748 . . . . . . 7 (𝑥 ∈ (𝑋𝐴) → 𝑥𝑋)
132psref 17409 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝑥𝑋) → 𝑥𝑅𝑥)
1412, 13sylan2 580 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥𝑅𝑥)
15 brinxp2 5318 . . . . . 6 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥𝐴𝑥𝐴𝑥𝑅𝑥))
1610, 10, 14, 15syl3anbrc 1428 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
17 vex 3354 . . . . . 6 𝑥 ∈ V
1817, 17breldm 5465 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)))
1916, 18syl 17 . . . 4 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)))
2019ex 397 . . 3 (𝑅 ∈ PosetRel → (𝑥 ∈ (𝑋𝐴) → 𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))))
2120ssrdv 3758 . 2 (𝑅 ∈ PosetRel → (𝑋𝐴) ⊆ dom (𝑅 ∩ (𝐴 × 𝐴)))
227, 21eqssd 3769 1 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cin 3722  wss 3723   class class class wbr 4786   × cxp 5247  dom cdm 5249  PosetRelcps 17399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ps 17401
This theorem is referenced by:  psssdm  17417  ordtrest  21220
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