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Theorem tsrdir 18662
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 18645 . . . 4 (𝐴 ∈ TosetRel → 𝐴 ∈ PosetRel)
2 psrel 18627 . . . 4 (𝐴 ∈ PosetRel → Rel 𝐴)
31, 2syl 17 . . 3 (𝐴 ∈ TosetRel → Rel 𝐴)
4 psref2 18628 . . . . 5 (𝐴 ∈ PosetRel → (𝐴𝐴) = ( I ↾ 𝐴))
5 inss1 4245 . . . . 5 (𝐴𝐴) ⊆ 𝐴
64, 5eqsstrrdi 4051 . . . 4 (𝐴 ∈ PosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
71, 6syl 17 . . 3 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
83, 7jca 511 . 2 (𝐴 ∈ TosetRel → (Rel 𝐴 ∧ ( I ↾ 𝐴) ⊆ 𝐴))
9 pstr2 18629 . . . 4 (𝐴 ∈ PosetRel → (𝐴𝐴) ⊆ 𝐴)
101, 9syl 17 . . 3 (𝐴 ∈ TosetRel → (𝐴𝐴) ⊆ 𝐴)
11 psdmrn 18631 . . . . . . 7 (𝐴 ∈ PosetRel → (dom 𝐴 = 𝐴 ∧ ran 𝐴 = 𝐴))
121, 11syl 17 . . . . . 6 (𝐴 ∈ TosetRel → (dom 𝐴 = 𝐴 ∧ ran 𝐴 = 𝐴))
1312simpld 494 . . . . 5 (𝐴 ∈ TosetRel → dom 𝐴 = 𝐴)
1413sqxpeqd 5721 . . . 4 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) = ( 𝐴 × 𝐴))
15 eqid 2735 . . . . . . 7 dom 𝐴 = dom 𝐴
1615istsr 18641 . . . . . 6 (𝐴 ∈ TosetRel ↔ (𝐴 ∈ PosetRel ∧ (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴)))
1716simprbi 496 . . . . 5 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴))
18 relcoi2 6299 . . . . . . . 8 (Rel 𝐴 → (( I ↾ 𝐴) ∘ 𝐴) = 𝐴)
193, 18syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (( I ↾ 𝐴) ∘ 𝐴) = 𝐴)
20 cnvresid 6647 . . . . . . . . 9 ( I ↾ 𝐴) = ( I ↾ 𝐴)
21 cnvss 5886 . . . . . . . . . 10 (( I ↾ 𝐴) ⊆ 𝐴( I ↾ 𝐴) ⊆ 𝐴)
227, 21syl 17 . . . . . . . . 9 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
2320, 22eqsstrrid 4045 . . . . . . . 8 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
24 coss1 5869 . . . . . . . 8 (( I ↾ 𝐴) ⊆ 𝐴 → (( I ↾ 𝐴) ∘ 𝐴) ⊆ (𝐴𝐴))
2523, 24syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (( I ↾ 𝐴) ∘ 𝐴) ⊆ (𝐴𝐴))
2619, 25eqsstrrd 4035 . . . . . 6 (𝐴 ∈ TosetRel → 𝐴 ⊆ (𝐴𝐴))
27 relcnv 6125 . . . . . . . 8 Rel 𝐴
28 relcoi1 6300 . . . . . . . 8 (Rel 𝐴 → (𝐴 ∘ ( I ↾ 𝐴)) = 𝐴)
2927, 28ax-mp 5 . . . . . . 7 (𝐴 ∘ ( I ↾ 𝐴)) = 𝐴
30 relcnvfld 6302 . . . . . . . . . . 11 (Rel 𝐴 𝐴 = 𝐴)
313, 30syl 17 . . . . . . . . . 10 (𝐴 ∈ TosetRel → 𝐴 = 𝐴)
3231reseq2d 6000 . . . . . . . . 9 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) = ( I ↾ 𝐴))
3332, 7eqsstrrd 4035 . . . . . . . 8 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
34 coss2 5870 . . . . . . . 8 (( I ↾ 𝐴) ⊆ 𝐴 → (𝐴 ∘ ( I ↾ 𝐴)) ⊆ (𝐴𝐴))
3533, 34syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (𝐴 ∘ ( I ↾ 𝐴)) ⊆ (𝐴𝐴))
3629, 35eqsstrrid 4045 . . . . . 6 (𝐴 ∈ TosetRel → 𝐴 ⊆ (𝐴𝐴))
3726, 36unssd 4202 . . . . 5 (𝐴 ∈ TosetRel → (𝐴𝐴) ⊆ (𝐴𝐴))
3817, 37sstrd 4006 . . . 4 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴))
3914, 38eqsstrrd 4035 . . 3 (𝐴 ∈ TosetRel → ( 𝐴 × 𝐴) ⊆ (𝐴𝐴))
4010, 39jca 511 . 2 (𝐴 ∈ TosetRel → ((𝐴𝐴) ⊆ 𝐴 ∧ ( 𝐴 × 𝐴) ⊆ (𝐴𝐴)))
41 eqid 2735 . . 3 𝐴 = 𝐴
4241isdir 18656 . 2 (𝐴 ∈ TosetRel → (𝐴 ∈ DirRel ↔ ((Rel 𝐴 ∧ ( I ↾ 𝐴) ⊆ 𝐴) ∧ ((𝐴𝐴) ⊆ 𝐴 ∧ ( 𝐴 × 𝐴) ⊆ (𝐴𝐴)))))
438, 40, 42mpbir2and 713 1 (𝐴 ∈ TosetRel → 𝐴 ∈ DirRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cun 3961  cin 3962  wss 3963   cuni 4912   I cid 5582   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  ccom 5693  Rel wrel 5694  PosetRelcps 18622   TosetRel ctsr 18623  DirRelcdir 18652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-ps 18624  df-tsr 18625  df-dir 18654
This theorem is referenced by: (None)
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