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Theorem tsrdir 18322
Description: A totally ordered set is a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
Assertion
Ref Expression
tsrdir (𝐴 ∈ TosetRel → 𝐴 ∈ DirRel)

Proof of Theorem tsrdir
StepHypRef Expression
1 tsrps 18305 . . . 4 (𝐴 ∈ TosetRel → 𝐴 ∈ PosetRel)
2 psrel 18287 . . . 4 (𝐴 ∈ PosetRel → Rel 𝐴)
31, 2syl 17 . . 3 (𝐴 ∈ TosetRel → Rel 𝐴)
4 psref2 18288 . . . . 5 (𝐴 ∈ PosetRel → (𝐴𝐴) = ( I ↾ 𝐴))
5 inss1 4162 . . . . 5 (𝐴𝐴) ⊆ 𝐴
64, 5eqsstrrdi 3976 . . . 4 (𝐴 ∈ PosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
71, 6syl 17 . . 3 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
83, 7jca 512 . 2 (𝐴 ∈ TosetRel → (Rel 𝐴 ∧ ( I ↾ 𝐴) ⊆ 𝐴))
9 pstr2 18289 . . . 4 (𝐴 ∈ PosetRel → (𝐴𝐴) ⊆ 𝐴)
101, 9syl 17 . . 3 (𝐴 ∈ TosetRel → (𝐴𝐴) ⊆ 𝐴)
11 psdmrn 18291 . . . . . . 7 (𝐴 ∈ PosetRel → (dom 𝐴 = 𝐴 ∧ ran 𝐴 = 𝐴))
121, 11syl 17 . . . . . 6 (𝐴 ∈ TosetRel → (dom 𝐴 = 𝐴 ∧ ran 𝐴 = 𝐴))
1312simpld 495 . . . . 5 (𝐴 ∈ TosetRel → dom 𝐴 = 𝐴)
1413sqxpeqd 5621 . . . 4 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) = ( 𝐴 × 𝐴))
15 eqid 2738 . . . . . . 7 dom 𝐴 = dom 𝐴
1615istsr 18301 . . . . . 6 (𝐴 ∈ TosetRel ↔ (𝐴 ∈ PosetRel ∧ (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴)))
1716simprbi 497 . . . . 5 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴))
18 relcoi2 6180 . . . . . . . 8 (Rel 𝐴 → (( I ↾ 𝐴) ∘ 𝐴) = 𝐴)
193, 18syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (( I ↾ 𝐴) ∘ 𝐴) = 𝐴)
20 cnvresid 6513 . . . . . . . . 9 ( I ↾ 𝐴) = ( I ↾ 𝐴)
21 cnvss 5781 . . . . . . . . . 10 (( I ↾ 𝐴) ⊆ 𝐴( I ↾ 𝐴) ⊆ 𝐴)
227, 21syl 17 . . . . . . . . 9 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
2320, 22eqsstrrid 3970 . . . . . . . 8 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
24 coss1 5764 . . . . . . . 8 (( I ↾ 𝐴) ⊆ 𝐴 → (( I ↾ 𝐴) ∘ 𝐴) ⊆ (𝐴𝐴))
2523, 24syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (( I ↾ 𝐴) ∘ 𝐴) ⊆ (𝐴𝐴))
2619, 25eqsstrrd 3960 . . . . . 6 (𝐴 ∈ TosetRel → 𝐴 ⊆ (𝐴𝐴))
27 relcnv 6012 . . . . . . . 8 Rel 𝐴
28 relcoi1 6181 . . . . . . . 8 (Rel 𝐴 → (𝐴 ∘ ( I ↾ 𝐴)) = 𝐴)
2927, 28ax-mp 5 . . . . . . 7 (𝐴 ∘ ( I ↾ 𝐴)) = 𝐴
30 relcnvfld 6183 . . . . . . . . . . 11 (Rel 𝐴 𝐴 = 𝐴)
313, 30syl 17 . . . . . . . . . 10 (𝐴 ∈ TosetRel → 𝐴 = 𝐴)
3231reseq2d 5891 . . . . . . . . 9 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) = ( I ↾ 𝐴))
3332, 7eqsstrrd 3960 . . . . . . . 8 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
34 coss2 5765 . . . . . . . 8 (( I ↾ 𝐴) ⊆ 𝐴 → (𝐴 ∘ ( I ↾ 𝐴)) ⊆ (𝐴𝐴))
3533, 34syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (𝐴 ∘ ( I ↾ 𝐴)) ⊆ (𝐴𝐴))
3629, 35eqsstrrid 3970 . . . . . 6 (𝐴 ∈ TosetRel → 𝐴 ⊆ (𝐴𝐴))
3726, 36unssd 4120 . . . . 5 (𝐴 ∈ TosetRel → (𝐴𝐴) ⊆ (𝐴𝐴))
3817, 37sstrd 3931 . . . 4 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴))
3914, 38eqsstrrd 3960 . . 3 (𝐴 ∈ TosetRel → ( 𝐴 × 𝐴) ⊆ (𝐴𝐴))
4010, 39jca 512 . 2 (𝐴 ∈ TosetRel → ((𝐴𝐴) ⊆ 𝐴 ∧ ( 𝐴 × 𝐴) ⊆ (𝐴𝐴)))
41 eqid 2738 . . 3 𝐴 = 𝐴
4241isdir 18316 . 2 (𝐴 ∈ TosetRel → (𝐴 ∈ DirRel ↔ ((Rel 𝐴 ∧ ( I ↾ 𝐴) ⊆ 𝐴) ∧ ((𝐴𝐴) ⊆ 𝐴 ∧ ( 𝐴 × 𝐴) ⊆ (𝐴𝐴)))))
438, 40, 42mpbir2and 710 1 (𝐴 ∈ TosetRel → 𝐴 ∈ DirRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  cun 3885  cin 3886  wss 3887   cuni 4839   I cid 5488   × cxp 5587  ccnv 5588  dom cdm 5589  ran crn 5590  cres 5591  ccom 5593  Rel wrel 5594  PosetRelcps 18282   TosetRel ctsr 18283  DirRelcdir 18312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-fun 6435  df-ps 18284  df-tsr 18285  df-dir 18314
This theorem is referenced by: (None)
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