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Theorem psss 17820
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 4158 . . 3 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
2 psrel 17809 . . 3 (𝑅 ∈ PosetRel → Rel 𝑅)
3 relss 5624 . . 3 ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 → (Rel 𝑅 → Rel (𝑅 ∩ (𝐴 × 𝐴))))
41, 2, 3mpsyl 68 . 2 (𝑅 ∈ PosetRel → Rel (𝑅 ∩ (𝐴 × 𝐴)))
5 pstr2 17811 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
6 trinxp 5956 . . 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 4827 . . . . . 6 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
98unissi 4812 . . . . 5 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
10 uniin 4827 . . . . 5 ( 𝑅 (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
119, 10sstri 3927 . . . 4 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
12 elin 3900 . . . . . 6 (𝑥 ∈ ( 𝑅 (𝐴 × 𝐴)) ↔ (𝑥 𝑅𝑥 (𝐴 × 𝐴)))
13 unixpid 6107 . . . . . . . . 9 (𝐴 × 𝐴) = 𝐴
1413eleq2i 2884 . . . . . . . 8 (𝑥 (𝐴 × 𝐴) ↔ 𝑥𝐴)
15 simprr 772 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥𝐴)
16 psdmrn 17813 . . . . . . . . . . . . . . 15 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
1716simpld 498 . . . . . . . . . . . . . 14 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
1817eleq2d 2878 . . . . . . . . . . . . 13 (𝑅 ∈ PosetRel → (𝑥 ∈ dom 𝑅𝑥 𝑅))
1918biimpar 481 . . . . . . . . . . . 12 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → 𝑥 ∈ dom 𝑅)
20 eqid 2801 . . . . . . . . . . . . 13 dom 𝑅 = dom 𝑅
2120psref 17814 . . . . . . . . . . . 12 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
2219, 21syldan 594 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → 𝑥𝑅𝑥)
2322adantrr 716 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥𝑅𝑥)
24 brinxp2 5597 . . . . . . . . . 10 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ ((𝑥𝐴𝑥𝐴) ∧ 𝑥𝑅𝑥))
2515, 15, 23, 24syl21anbrc 1341 . . . . . . . . 9 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
2625expr 460 . . . . . . . 8 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → (𝑥𝐴𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2714, 26syl5bi 245 . . . . . . 7 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → (𝑥 (𝐴 × 𝐴) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2827expimpd 457 . . . . . 6 (𝑅 ∈ PosetRel → ((𝑥 𝑅𝑥 (𝐴 × 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2912, 28syl5bi 245 . . . . 5 (𝑅 ∈ PosetRel → (𝑥 ∈ ( 𝑅 (𝐴 × 𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3029ralrimiv 3151 . . . 4 (𝑅 ∈ PosetRel → ∀𝑥 ∈ ( 𝑅 (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
31 ssralv 3984 . . . 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 5078 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥𝑅𝑦)
341ssbri 5078 . . . . 5 (𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥𝑦𝑅𝑥)
35 psasym 17816 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)
36353expib 1119 . . . . 5 (𝑅 ∈ PosetRel → ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3733, 34, 36syl2ani 609 . . . 4 (𝑅 ∈ PosetRel → ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
3837alrimivv 1929 . . 3 (𝑅 ∈ PosetRel → ∀𝑥𝑦((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
39 asymref2 5948 . . 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 586 . 2 (𝑅 ∈ PosetRel → ((𝑅 ∩ (𝐴 × 𝐴)) ∩ (𝑅 ∩ (𝐴 × 𝐴))) = ( I ↾ (𝑅 ∩ (𝐴 × 𝐴))))
41 inex1g 5190 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
42 isps 17808 . . 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 1339 1 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wal 1536   = wceq 1538  wcel 2112  wral 3109  Vcvv 3444  cin 3883  wss 3884   cuni 4803   class class class wbr 5033   I cid 5427   × cxp 5521  ccnv 5522  dom cdm 5523  ran crn 5524  cres 5525  ccom 5527  Rel wrel 5528  PosetRelcps 17804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ps 17806
This theorem is referenced by:  tsrss  17829  ordtrest2  21813
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