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Theorem tsrdir 17007
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 16990 . . . 4 (𝐴 ∈ TosetRel → 𝐴 ∈ PosetRel)
2 psrel 16972 . . . 4 (𝐴 ∈ PosetRel → Rel 𝐴)
31, 2syl 17 . . 3 (𝐴 ∈ TosetRel → Rel 𝐴)
4 psref2 16973 . . . . 5 (𝐴 ∈ PosetRel → (𝐴𝐴) = ( I ↾ 𝐴))
5 inss1 3794 . . . . 5 (𝐴𝐴) ⊆ 𝐴
64, 5syl6eqssr 3618 . . . 4 (𝐴 ∈ PosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
71, 6syl 17 . . 3 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
83, 7jca 552 . 2 (𝐴 ∈ TosetRel → (Rel 𝐴 ∧ ( I ↾ 𝐴) ⊆ 𝐴))
9 pstr2 16974 . . . 4 (𝐴 ∈ PosetRel → (𝐴𝐴) ⊆ 𝐴)
101, 9syl 17 . . 3 (𝐴 ∈ TosetRel → (𝐴𝐴) ⊆ 𝐴)
11 psdmrn 16976 . . . . . . 7 (𝐴 ∈ PosetRel → (dom 𝐴 = 𝐴 ∧ ran 𝐴 = 𝐴))
121, 11syl 17 . . . . . 6 (𝐴 ∈ TosetRel → (dom 𝐴 = 𝐴 ∧ ran 𝐴 = 𝐴))
1312simpld 473 . . . . 5 (𝐴 ∈ TosetRel → dom 𝐴 = 𝐴)
1413sqxpeqd 5055 . . . 4 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) = ( 𝐴 × 𝐴))
15 eqid 2609 . . . . . . 7 dom 𝐴 = dom 𝐴
1615istsr 16986 . . . . . 6 (𝐴 ∈ TosetRel ↔ (𝐴 ∈ PosetRel ∧ (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴)))
1716simprbi 478 . . . . 5 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴))
18 relcoi2 5566 . . . . . . . 8 (Rel 𝐴 → (( I ↾ 𝐴) ∘ 𝐴) = 𝐴)
193, 18syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (( I ↾ 𝐴) ∘ 𝐴) = 𝐴)
20 cnvresid 5868 . . . . . . . . 9 ( I ↾ 𝐴) = ( I ↾ 𝐴)
21 cnvss 5204 . . . . . . . . . 10 (( I ↾ 𝐴) ⊆ 𝐴( I ↾ 𝐴) ⊆ 𝐴)
227, 21syl 17 . . . . . . . . 9 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
2320, 22syl5eqssr 3612 . . . . . . . 8 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
24 coss1 5187 . . . . . . . 8 (( I ↾ 𝐴) ⊆ 𝐴 → (( I ↾ 𝐴) ∘ 𝐴) ⊆ (𝐴𝐴))
2523, 24syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (( I ↾ 𝐴) ∘ 𝐴) ⊆ (𝐴𝐴))
2619, 25eqsstr3d 3602 . . . . . 6 (𝐴 ∈ TosetRel → 𝐴 ⊆ (𝐴𝐴))
27 relcnv 5409 . . . . . . . 8 Rel 𝐴
28 relcoi1 5567 . . . . . . . 8 (Rel 𝐴 → (𝐴 ∘ ( I ↾ 𝐴)) = 𝐴)
2927, 28ax-mp 5 . . . . . . 7 (𝐴 ∘ ( I ↾ 𝐴)) = 𝐴
30 relcnvfld 5569 . . . . . . . . . . 11 (Rel 𝐴 𝐴 = 𝐴)
313, 30syl 17 . . . . . . . . . 10 (𝐴 ∈ TosetRel → 𝐴 = 𝐴)
3231reseq2d 5304 . . . . . . . . 9 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) = ( I ↾ 𝐴))
3332, 7eqsstr3d 3602 . . . . . . . 8 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
34 coss2 5188 . . . . . . . 8 (( I ↾ 𝐴) ⊆ 𝐴 → (𝐴 ∘ ( I ↾ 𝐴)) ⊆ (𝐴𝐴))
3533, 34syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (𝐴 ∘ ( I ↾ 𝐴)) ⊆ (𝐴𝐴))
3629, 35syl5eqssr 3612 . . . . . 6 (𝐴 ∈ TosetRel → 𝐴 ⊆ (𝐴𝐴))
3726, 36unssd 3750 . . . . 5 (𝐴 ∈ TosetRel → (𝐴𝐴) ⊆ (𝐴𝐴))
3817, 37sstrd 3577 . . . 4 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴))
3914, 38eqsstr3d 3602 . . 3 (𝐴 ∈ TosetRel → ( 𝐴 × 𝐴) ⊆ (𝐴𝐴))
4010, 39jca 552 . 2 (𝐴 ∈ TosetRel → ((𝐴𝐴) ⊆ 𝐴 ∧ ( 𝐴 × 𝐴) ⊆ (𝐴𝐴)))
41 eqid 2609 . . 3 𝐴 = 𝐴
4241isdir 17001 . 2 (𝐴 ∈ TosetRel → (𝐴 ∈ DirRel ↔ ((Rel 𝐴 ∧ ( I ↾ 𝐴) ⊆ 𝐴) ∧ ((𝐴𝐴) ⊆ 𝐴 ∧ ( 𝐴 × 𝐴) ⊆ (𝐴𝐴)))))
438, 40, 42mpbir2and 958 1 (𝐴 ∈ TosetRel → 𝐴 ∈ DirRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cun 3537  cin 3538  wss 3539   cuni 4366   I cid 4938   × cxp 5026  ccnv 5027  dom cdm 5028  ran crn 5029  cres 5030  ccom 5032  Rel wrel 5033  PosetRelcps 16967   TosetRel ctsr 16968  DirRelcdir 16997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-fun 5792  df-ps 16969  df-tsr 16970  df-dir 16999
This theorem is referenced by: (None)
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