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Theorem psss 17135
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 3811 . . 3 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
2 psrel 17124 . . 3 (𝑅 ∈ PosetRel → Rel 𝑅)
3 relss 5167 . . 3 ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 → (Rel 𝑅 → Rel (𝑅 ∩ (𝐴 × 𝐴))))
41, 2, 3mpsyl 68 . 2 (𝑅 ∈ PosetRel → Rel (𝑅 ∩ (𝐴 × 𝐴)))
5 pstr2 17126 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
6 trinxp 5480 . . 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 4423 . . . . . 6 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
98unissi 4427 . . . . 5 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
10 uniin 4423 . . . . 5 ( 𝑅 (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
119, 10sstri 3592 . . . 4 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
12 elin 3774 . . . . . 6 (𝑥 ∈ ( 𝑅 (𝐴 × 𝐴)) ↔ (𝑥 𝑅𝑥 (𝐴 × 𝐴)))
13 unixpid 5629 . . . . . . . . 9 (𝐴 × 𝐴) = 𝐴
1413eleq2i 2690 . . . . . . . 8 (𝑥 (𝐴 × 𝐴) ↔ 𝑥𝐴)
15 simprr 795 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥𝐴)
16 psdmrn 17128 . . . . . . . . . . . . . . 15 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
1716simpld 475 . . . . . . . . . . . . . 14 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
1817eleq2d 2684 . . . . . . . . . . . . 13 (𝑅 ∈ PosetRel → (𝑥 ∈ dom 𝑅𝑥 𝑅))
1918biimpar 502 . . . . . . . . . . . 12 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → 𝑥 ∈ dom 𝑅)
20 eqid 2621 . . . . . . . . . . . . 13 dom 𝑅 = dom 𝑅
2120psref 17129 . . . . . . . . . . . 12 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
2219, 21syldan 487 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → 𝑥𝑅𝑥)
2322adantrr 752 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥𝑅𝑥)
24 brinxp2 5141 . . . . . . . . . 10 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥𝐴𝑥𝐴𝑥𝑅𝑥))
2515, 15, 23, 24syl3anbrc 1244 . . . . . . . . 9 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
2625expr 642 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → (𝑥𝐴𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2714, 26syl5bi 232 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → (𝑥 (𝐴 × 𝐴) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2827expimpd 628 . . . . . 6 (𝑅 ∈ PosetRel → ((𝑥 𝑅𝑥 (𝐴 × 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2912, 28syl5bi 232 . . . . 5 (𝑅 ∈ PosetRel → (𝑥 ∈ ( 𝑅 (𝐴 × 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3029ralrimiv 2959 . . . 4 (𝑅 ∈ PosetRel → ∀𝑥 ∈ ( 𝑅 (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
31 ssralv 3645 . . . 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 4657 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥𝑅𝑦)
341ssbri 4657 . . . . 5 (𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥𝑦𝑅𝑥)
35 psasym 17131 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)
36353expib 1265 . . . . 5 (𝑅 ∈ PosetRel → ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3733, 34, 36syl2ani 687 . . . 4 (𝑅 ∈ PosetRel → ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
3837alrimivv 1853 . . 3 (𝑅 ∈ PosetRel → ∀𝑥𝑦((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
39 asymref2 5472 . . 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 697 . 2 (𝑅 ∈ PosetRel → ((𝑅 ∩ (𝐴 × 𝐴)) ∩ (𝑅 ∩ (𝐴 × 𝐴))) = ( I ↾ (𝑅 ∩ (𝐴 × 𝐴))))
41 inex1g 4761 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
42 isps 17123 . . 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 1243 1 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cin 3554  wss 3555   cuni 4402   class class class wbr 4613   I cid 4984   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  cres 5076  ccom 5078  Rel wrel 5079  PosetRelcps 17119
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 4741  ax-nul 4749  ax-pr 4867
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 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ps 17121
This theorem is referenced by:  tsrss  17144  ordtrest2  20918
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