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Theorem psss 17336
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 3941 . . 3 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
2 psrel 17325 . . 3 (𝑅 ∈ PosetRel → Rel 𝑅)
3 relss 5315 . . 3 ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 → (Rel 𝑅 → Rel (𝑅 ∩ (𝐴 × 𝐴))))
41, 2, 3mpsyl 68 . 2 (𝑅 ∈ PosetRel → Rel (𝑅 ∩ (𝐴 × 𝐴)))
5 pstr2 17327 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
6 trinxp 5631 . . 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 4565 . . . . . 6 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
98unissi 4569 . . . . 5 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
10 uniin 4565 . . . . 5 ( 𝑅 (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
119, 10sstri 3718 . . . 4 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ ( 𝑅 (𝐴 × 𝐴))
12 elin 3904 . . . . . 6 (𝑥 ∈ ( 𝑅 (𝐴 × 𝐴)) ↔ (𝑥 𝑅𝑥 (𝐴 × 𝐴)))
13 unixpid 5783 . . . . . . . . 9 (𝐴 × 𝐴) = 𝐴
1413eleq2i 2795 . . . . . . . 8 (𝑥 (𝐴 × 𝐴) ↔ 𝑥𝐴)
15 simprr 813 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥𝐴)
16 psdmrn 17329 . . . . . . . . . . . . . . 15 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
1716simpld 477 . . . . . . . . . . . . . 14 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
1817eleq2d 2789 . . . . . . . . . . . . 13 (𝑅 ∈ PosetRel → (𝑥 ∈ dom 𝑅𝑥 𝑅))
1918biimpar 503 . . . . . . . . . . . 12 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → 𝑥 ∈ dom 𝑅)
20 eqid 2724 . . . . . . . . . . . . 13 dom 𝑅 = dom 𝑅
2120psref 17330 . . . . . . . . . . . 12 ((𝑅 ∈ PosetRel ∧ 𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
2219, 21syldan 488 . . . . . . . . . . 11 ((𝑅 ∈ PosetRel ∧ 𝑥 𝑅) → 𝑥𝑅𝑥)
2322adantrr 755 . . . . . . . . . 10 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥𝑅𝑥)
24 brinxp2 5289 . . . . . . . . . 10 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥 ↔ (𝑥𝐴𝑥𝐴𝑥𝑅𝑥))
2515, 15, 23, 24syl3anbrc 1383 . . . . . . . . 9 ((𝑅 ∈ PosetRel ∧ (𝑥 𝑅𝑥𝐴)) → 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
2625expr 644 . . . . . . . 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 630 . . . . . 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 3067 . . . 4 (𝑅 ∈ PosetRel → ∀𝑥 ∈ ( 𝑅 (𝐴 × 𝐴))𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥)
31 ssralv 3772 . . . 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 4805 . . . . 5 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑥𝑅𝑦)
341ssbri 4805 . . . . 5 (𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥𝑦𝑅𝑥)
35 psasym 17332 . . . . . 6 ((𝑅 ∈ PosetRel ∧ 𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦)
36353expib 1116 . . . . 5 (𝑅 ∈ PosetRel → ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
3733, 34, 36syl2ani 691 . . . 4 (𝑅 ∈ PosetRel → ((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
3837alrimivv 1969 . . 3 (𝑅 ∈ PosetRel → ∀𝑥𝑦((𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥) → 𝑥 = 𝑦))
39 asymref2 5623 . . 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 701 . 2 (𝑅 ∈ PosetRel → ((𝑅 ∩ (𝐴 × 𝐴)) ∩ (𝑅 ∩ (𝐴 × 𝐴))) = ( I ↾ (𝑅 ∩ (𝐴 × 𝐴))))
41 inex1g 4909 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
42 isps 17324 . . 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 1382 1 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wal 1594   = wceq 1596  wcel 2103  wral 3014  Vcvv 3304  cin 3679  wss 3680   cuni 4544   class class class wbr 4760   I cid 5127   × cxp 5216  ccnv 5217  dom cdm 5218  ran crn 5219  cres 5220  ccom 5222  Rel wrel 5223  PosetRelcps 17320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ps 17322
This theorem is referenced by:  tsrss  17345  ordtrest2  21131
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