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Theorem psss 17694
Description: Any subset of a partially ordered set is partially ordered. (Contributed by FL, 24-Jan-2010.)
Assertion
Ref Expression
psss (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)

Proof of Theorem psss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 4095 . . 3 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
2 psrel 17683 . . 3 (𝑅 ∈ PosetRel → Rel 𝑅)
3 relss 5510 . . 3 ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 → (Rel 𝑅 → Rel (𝑅 ∩ (𝐴 × 𝐴))))
41, 2, 3mpsyl 68 . 2 (𝑅 ∈ PosetRel → Rel (𝑅 ∩ (𝐴 × 𝐴)))
5 pstr2 17685 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
6 trinxp 5830 . . 3 ((𝑅𝑅) ⊆ 𝑅 → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
75, 6syl 17 . 2 (𝑅 ∈ PosetRel → ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴)))
8 uniin 4737 . . . . . 6 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
98unissi 4740 . . . . 5 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
10 uniin 4737 . . . . 5 ( 𝑅 (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
119, 10sstri 3869 . . . 4 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
12 elin 4060 . . . . . 6 (𝑥 ∈ ( 𝑅 (𝐴 × 𝐴)) ↔ (𝑥 𝑅𝑥 (𝐴 × 𝐴)))
13 unixpid 5978 . . . . . . . . 9 (𝐴 × 𝐴) = 𝐴
1413eleq2i 2859 . . . . . . . 8 (𝑥 (𝐴 × 𝐴) ↔ 𝑥𝐴)
15 simprr 761 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥𝐴)
16 psdmrn 17687 . . . . . . . . . . . . . . 15 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
1716simpld 487 . . . . . . . . . . . . . 14 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
1817eleq2d 2853 . . . . . . . . . . . . 13 (𝑅 ∈ PosetRel → (𝑥 ∈ dom 𝑅𝑥 𝑅))
1918biimpar 470 . . . . . . . . . . . 12 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → 𝑥 ∈ dom 𝑅)
20 eqid 2780 . . . . . . . . . . . . 13 dom 𝑅 = dom 𝑅
2120psref 17688 . . . . . . . . . . . 12 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
2219, 21syldan 583 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → 𝑥𝑅𝑥)
2322adantrr 705 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥𝑅𝑥)
24 brinxp2 5483 . . . . . . . . . 10 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥𝐴𝑥𝐴) ∧ 𝑥𝑅𝑥))
2515, 15, 23, 24syl21anbrc 1325 . . . . . . . . 9 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
2625expr 449 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → (𝑥𝐴𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2714, 26syl5bi 234 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → (𝑥 (𝐴 × 𝐴) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2827expimpd 446 . . . . . 6 (𝑅 ∈ PosetRel → ((𝑥 𝑅𝑥 (𝐴 × 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2912, 28syl5bi 234 . . . . 5 (𝑅 ∈ PosetRel → (𝑥 ∈ ( 𝑅 (𝐴 × 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3029ralrimiv 3133 . . . 4 (𝑅 ∈ PosetRel → ∀𝑥 ∈ ( 𝑅 (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
31 ssralv 3925 . . . 4 ( (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴)) → (∀𝑥 ∈ ( 𝑅 (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 → ∀𝑥 (𝑅 ∩ (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3211, 30, 31mpsyl 68 . . 3 (𝑅 ∈ PosetRel → ∀𝑥 (𝑅 ∩ (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
331ssbri 4979 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥𝑅𝑦)
341ssbri 4979 . . . . 5 (𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥𝑦𝑅𝑥)
35 psasym 17690 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)
36353expib 1103 . . . . 5 (𝑅 ∈ PosetRel → ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3733, 34, 36syl2ani 598 . . . 4 (𝑅 ∈ PosetRel → ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
3837alrimivv 1888 . . 3 (𝑅 ∈ PosetRel → ∀𝑥𝑦((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
39 asymref2 5822 . . 3 (((𝑅 ∩ (𝐴 × 𝐴)) ∩ (𝑅 ∩ (𝐴 × 𝐴))) = ( I ↾ (𝑅 ∩ (𝐴 × 𝐴))) ↔ (∀𝑥 (𝑅 ∩ (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ∧ ∀𝑥𝑦((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦)))
4032, 38, 39sylanbrc 575 . 2 (𝑅 ∈ PosetRel → ((𝑅 ∩ (𝐴 × 𝐴)) ∩ (𝑅 ∩ (𝐴 × 𝐴))) = ( I ↾ (𝑅 ∩ (𝐴 × 𝐴))))
41 inex1g 5084 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
42 isps 17682 . . 3 ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel ↔ (Rel (𝑅 ∩ (𝐴 × 𝐴)) ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∩ (𝑅 ∩ (𝐴 × 𝐴))) = ( I ↾ (𝑅 ∩ (𝐴 × 𝐴))))))
4341, 42syl 17 . 2 (𝑅 ∈ PosetRel → ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel ↔ (Rel (𝑅 ∩ (𝐴 × 𝐴)) ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∩ (𝑅 ∩ (𝐴 × 𝐴))) = ( I ↾ (𝑅 ∩ (𝐴 × 𝐴))))))
444, 7, 40, 43mpbir3and 1323 1 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069  wal 1506   = wceq 1508  wcel 2051  wral 3090  Vcvv 3417  cin 3830  wss 3831   cuni 4717   class class class wbr 4934   I cid 5315   × cxp 5409  ccnv 5410  dom cdm 5411  ran crn 5412  cres 5413  ccom 5415  Rel wrel 5416  PosetRelcps 17678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-sep 5064  ax-nul 5071  ax-pr 5190
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3419  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-op 4451  df-uni 4718  df-br 4935  df-opab 4997  df-id 5316  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ps 17680
This theorem is referenced by:  tsrss  17703  ordtrest2  21531
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