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Theorem psssdm2 17482
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 5499 . . . 4 dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (dom 𝑅 ∩ dom (𝐴 × 𝐴))
2 psssdm.1 . . . . . 6 𝑋 = dom 𝑅
32eqcomi 2773 . . . . 5 dom 𝑅 = 𝑋
4 dmxpid 5512 . . . . 5 dom (𝐴 × 𝐴) = 𝐴
53, 4ineq12i 3973 . . . 4 (dom 𝑅 ∩ dom (𝐴 × 𝐴)) = (𝑋𝐴)
61, 5sseqtri 3796 . . 3 dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋𝐴)
76a1i 11 . 2 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋𝐴))
8 simpr 477 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥 ∈ (𝑋𝐴))
98elin2d 3964 . . . 4 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥𝐴)
10 elinel1 3960 . . . . 5 (𝑥 ∈ (𝑋𝐴) → 𝑥𝑋)
112psref 17475 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝑥𝑋) → 𝑥𝑅𝑥)
1210, 11sylan2 586 . . . 4 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥𝑅𝑥)
13 brinxp2 5347 . . . 4 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥𝐴𝑥𝐴) ∧ 𝑥𝑅𝑥))
149, 9, 12, 13syl21anbrc 1444 . . 3 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
15 vex 3352 . . . 4 𝑥 ∈ V
1615, 15breldm 5496 . . 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 3781 1 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  cin 3730  wss 3731   class class class wbr 4808   × cxp 5274  dom cdm 5276  PosetRelcps 17465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-br 4809  df-opab 4871  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ps 17467
This theorem is referenced by:  psssdm  17483  ordtrest  21285
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