Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  psssdm2 Structured version   Visualization version   GIF version

Theorem psssdm2 17884
 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 5752 . . . 4 dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (dom 𝑅 ∩ dom (𝐴 × 𝐴))
2 psssdm.1 . . . . . 6 𝑋 = dom 𝑅
32eqcomi 2768 . . . . 5 dom 𝑅 = 𝑋
4 dmxpid 5772 . . . . 5 dom (𝐴 × 𝐴) = 𝐴
53, 4ineq12i 4116 . . . 4 (dom 𝑅 ∩ dom (𝐴 × 𝐴)) = (𝑋𝐴)
61, 5sseqtri 3929 . . 3 dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋𝐴)
76a1i 11 . 2 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝑋𝐴))
8 simpr 489 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥 ∈ (𝑋𝐴))
98elin2d 4105 . . . 4 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥𝐴)
10 elinel1 4101 . . . . 5 (𝑥 ∈ (𝑋𝐴) → 𝑥𝑋)
112psref 17877 . . . . 5 ((𝑅 ∈ PosetRel ∧ 𝑥𝑋) → 𝑥𝑅𝑥)
1210, 11sylan2 596 . . . 4 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥𝑅𝑥)
13 brinxp2 5599 . . . 4 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥𝐴𝑥𝐴) ∧ 𝑥𝑅𝑥))
149, 9, 12, 13syl21anbrc 1342 . . 3 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ (𝑋𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
15 vex 3414 . . . 4 𝑥 ∈ V
1615, 15breldm 5749 . . 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 3914 1 (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ∩ cin 3858   ⊆ wss 3859   class class class wbr 5033   × cxp 5523  dom cdm 5525  PosetRelcps 17867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ps 17869 This theorem is referenced by:  psssdm  17885  ordtrest  21895
 Copyright terms: Public domain W3C validator