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Theorem psss 17422
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 3981 . . 3 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
2 psrel 17411 . . 3 (𝑅 ∈ PosetRel → Rel 𝑅)
3 relss 5346 . . 3 ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 → (Rel 𝑅 → Rel (𝑅 ∩ (𝐴 × 𝐴))))
41, 2, 3mpsyl 68 . 2 (𝑅 ∈ PosetRel → Rel (𝑅 ∩ (𝐴 × 𝐴)))
5 pstr2 17413 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
6 trinxp 5662 . . 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 4594 . . . . . 6 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
98unissi 4597 . . . . 5 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
10 uniin 4594 . . . . 5 ( 𝑅 (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
119, 10sstri 3761 . . . 4 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
12 elin 3947 . . . . . 6 (𝑥 ∈ ( 𝑅 (𝐴 × 𝐴)) ↔ (𝑥 𝑅𝑥 (𝐴 × 𝐴)))
13 unixpid 5814 . . . . . . . . 9 (𝐴 × 𝐴) = 𝐴
1413eleq2i 2842 . . . . . . . 8 (𝑥 (𝐴 × 𝐴) ↔ 𝑥𝐴)
15 simprr 756 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥𝐴)
16 psdmrn 17415 . . . . . . . . . . . . . . 15 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
1716simpld 482 . . . . . . . . . . . . . 14 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
1817eleq2d 2836 . . . . . . . . . . . . 13 (𝑅 ∈ PosetRel → (𝑥 ∈ dom 𝑅𝑥 𝑅))
1918biimpar 463 . . . . . . . . . . . 12 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → 𝑥 ∈ dom 𝑅)
20 eqid 2771 . . . . . . . . . . . . 13 dom 𝑅 = dom 𝑅
2120psref 17416 . . . . . . . . . . . 12 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
2219, 21syldan 579 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → 𝑥𝑅𝑥)
2322adantrr 696 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥𝑅𝑥)
24 brinxp2 5320 . . . . . . . . . 10 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥𝐴𝑥𝐴𝑥𝑅𝑥))
2515, 15, 23, 24syl3anbrc 1428 . . . . . . . . 9 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
2625expr 444 . . . . . . . 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 441 . . . . . 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 3114 . . . 4 (𝑅 ∈ PosetRel → ∀𝑥 ∈ ( 𝑅 (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
31 ssralv 3815 . . . 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 4831 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥𝑅𝑦)
341ssbri 4831 . . . . 5 (𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥𝑦𝑅𝑥)
35 psasym 17418 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)
36353expib 1116 . . . . 5 (𝑅 ∈ PosetRel → ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3733, 34, 36syl2ani 594 . . . 4 (𝑅 ∈ PosetRel → ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
3837alrimivv 2008 . . 3 (𝑅 ∈ PosetRel → ∀𝑥𝑦((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
39 asymref2 5654 . . 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 572 . 2 (𝑅 ∈ PosetRel → ((𝑅 ∩ (𝐴 × 𝐴)) ∩ (𝑅 ∩ (𝐴 × 𝐴))) = ( I ↾ (𝑅 ∩ (𝐴 × 𝐴))))
41 inex1g 4935 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
42 isps 17410 . . 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 1427 1 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071  wal 1629   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cin 3722  wss 3723   cuni 4574   class class class wbr 4786   I cid 5156   × cxp 5247  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251  ccom 5253  Rel wrel 5254  PosetRelcps 17406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ps 17408
This theorem is referenced by:  tsrss  17431  ordtrest2  21229
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