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Theorem tsrdir 17840
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 17823 . . . 4 (𝐴 ∈ TosetRel → 𝐴 ∈ PosetRel)
2 psrel 17805 . . . 4 (𝐴 ∈ PosetRel → Rel 𝐴)
31, 2syl 17 . . 3 (𝐴 ∈ TosetRel → Rel 𝐴)
4 psref2 17806 . . . . 5 (𝐴 ∈ PosetRel → (𝐴𝐴) = ( I ↾ 𝐴))
5 inss1 4208 . . . . 5 (𝐴𝐴) ⊆ 𝐴
64, 5eqsstrrdi 4025 . . . 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 17807 . . . 4 (𝐴 ∈ PosetRel → (𝐴𝐴) ⊆ 𝐴)
101, 9syl 17 . . 3 (𝐴 ∈ TosetRel → (𝐴𝐴) ⊆ 𝐴)
11 psdmrn 17809 . . . . . . 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 5585 . . . 4 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) = ( 𝐴 × 𝐴))
15 eqid 2825 . . . . . . 7 dom 𝐴 = dom 𝐴
1615istsr 17819 . . . . . 6 (𝐴 ∈ TosetRel ↔ (𝐴 ∈ PosetRel ∧ (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴)))
1716simprbi 497 . . . . 5 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴))
18 relcoi2 6125 . . . . . . . 8 (Rel 𝐴 → (( I ↾ 𝐴) ∘ 𝐴) = 𝐴)
193, 18syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (( I ↾ 𝐴) ∘ 𝐴) = 𝐴)
20 cnvresid 6429 . . . . . . . . 9 ( I ↾ 𝐴) = ( I ↾ 𝐴)
21 cnvss 5741 . . . . . . . . . 10 (( I ↾ 𝐴) ⊆ 𝐴( I ↾ 𝐴) ⊆ 𝐴)
227, 21syl 17 . . . . . . . . 9 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
2320, 22eqsstrrid 4019 . . . . . . . 8 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
24 coss1 5724 . . . . . . . 8 (( I ↾ 𝐴) ⊆ 𝐴 → (( I ↾ 𝐴) ∘ 𝐴) ⊆ (𝐴𝐴))
2523, 24syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (( I ↾ 𝐴) ∘ 𝐴) ⊆ (𝐴𝐴))
2619, 25eqsstrrd 4009 . . . . . 6 (𝐴 ∈ TosetRel → 𝐴 ⊆ (𝐴𝐴))
27 relcnv 5964 . . . . . . . 8 Rel 𝐴
28 relcoi1 6126 . . . . . . . 8 (Rel 𝐴 → (𝐴 ∘ ( I ↾ 𝐴)) = 𝐴)
2927, 28ax-mp 5 . . . . . . 7 (𝐴 ∘ ( I ↾ 𝐴)) = 𝐴
30 relcnvfld 6128 . . . . . . . . . . 11 (Rel 𝐴 𝐴 = 𝐴)
313, 30syl 17 . . . . . . . . . 10 (𝐴 ∈ TosetRel → 𝐴 = 𝐴)
3231reseq2d 5851 . . . . . . . . 9 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) = ( I ↾ 𝐴))
3332, 7eqsstrrd 4009 . . . . . . . 8 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
34 coss2 5725 . . . . . . . 8 (( I ↾ 𝐴) ⊆ 𝐴 → (𝐴 ∘ ( I ↾ 𝐴)) ⊆ (𝐴𝐴))
3533, 34syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (𝐴 ∘ ( I ↾ 𝐴)) ⊆ (𝐴𝐴))
3629, 35eqsstrrid 4019 . . . . . 6 (𝐴 ∈ TosetRel → 𝐴 ⊆ (𝐴𝐴))
3726, 36unssd 4165 . . . . 5 (𝐴 ∈ TosetRel → (𝐴𝐴) ⊆ (𝐴𝐴))
3817, 37sstrd 3980 . . . 4 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴))
3914, 38eqsstrrd 4009 . . 3 (𝐴 ∈ TosetRel → ( 𝐴 × 𝐴) ⊆ (𝐴𝐴))
4010, 39jca 512 . 2 (𝐴 ∈ TosetRel → ((𝐴𝐴) ⊆ 𝐴 ∧ ( 𝐴 × 𝐴) ⊆ (𝐴𝐴)))
41 eqid 2825 . . 3 𝐴 = 𝐴
4241isdir 17834 . 2 (𝐴 ∈ TosetRel → (𝐴 ∈ DirRel ↔ ((Rel 𝐴 ∧ ( I ↾ 𝐴) ⊆ 𝐴) ∧ ((𝐴𝐴) ⊆ 𝐴 ∧ ( 𝐴 × 𝐴) ⊆ (𝐴𝐴)))))
438, 40, 42mpbir2and 709 1 (𝐴 ∈ TosetRel → 𝐴 ∈ DirRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  cun 3937  cin 3938  wss 3939   cuni 4836   I cid 5457   × cxp 5551  ccnv 5552  dom cdm 5553  ran crn 5554  cres 5555  ccom 5557  Rel wrel 5558  PosetRelcps 17800   TosetRel ctsr 17801  DirRelcdir 17830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-fun 6353  df-ps 17802  df-tsr 17803  df-dir 17832
This theorem is referenced by: (None)
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