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Theorem tsrdir 17504
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 17487 . . . 4 (𝐴 ∈ TosetRel → 𝐴 ∈ PosetRel)
2 psrel 17469 . . . 4 (𝐴 ∈ PosetRel → Rel 𝐴)
31, 2syl 17 . . 3 (𝐴 ∈ TosetRel → Rel 𝐴)
4 psref2 17470 . . . . 5 (𝐴 ∈ PosetRel → (𝐴𝐴) = ( I ↾ 𝐴))
5 inss1 3992 . . . . 5 (𝐴𝐴) ⊆ 𝐴
64, 5syl6eqssr 3816 . . . 4 (𝐴 ∈ PosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
71, 6syl 17 . . 3 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
83, 7jca 507 . 2 (𝐴 ∈ TosetRel → (Rel 𝐴 ∧ ( I ↾ 𝐴) ⊆ 𝐴))
9 pstr2 17471 . . . 4 (𝐴 ∈ PosetRel → (𝐴𝐴) ⊆ 𝐴)
101, 9syl 17 . . 3 (𝐴 ∈ TosetRel → (𝐴𝐴) ⊆ 𝐴)
11 psdmrn 17473 . . . . . . 7 (𝐴 ∈ PosetRel → (dom 𝐴 = 𝐴 ∧ ran 𝐴 = 𝐴))
121, 11syl 17 . . . . . 6 (𝐴 ∈ TosetRel → (dom 𝐴 = 𝐴 ∧ ran 𝐴 = 𝐴))
1312simpld 488 . . . . 5 (𝐴 ∈ TosetRel → dom 𝐴 = 𝐴)
1413sqxpeqd 5309 . . . 4 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) = ( 𝐴 × 𝐴))
15 eqid 2765 . . . . . . 7 dom 𝐴 = dom 𝐴
1615istsr 17483 . . . . . 6 (𝐴 ∈ TosetRel ↔ (𝐴 ∈ PosetRel ∧ (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴)))
1716simprbi 490 . . . . 5 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴))
18 relcoi2 5849 . . . . . . . 8 (Rel 𝐴 → (( I ↾ 𝐴) ∘ 𝐴) = 𝐴)
193, 18syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (( I ↾ 𝐴) ∘ 𝐴) = 𝐴)
20 cnvresid 6146 . . . . . . . . 9 ( I ↾ 𝐴) = ( I ↾ 𝐴)
21 cnvss 5463 . . . . . . . . . 10 (( I ↾ 𝐴) ⊆ 𝐴( I ↾ 𝐴) ⊆ 𝐴)
227, 21syl 17 . . . . . . . . 9 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
2320, 22syl5eqssr 3810 . . . . . . . 8 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
24 coss1 5446 . . . . . . . 8 (( I ↾ 𝐴) ⊆ 𝐴 → (( I ↾ 𝐴) ∘ 𝐴) ⊆ (𝐴𝐴))
2523, 24syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (( I ↾ 𝐴) ∘ 𝐴) ⊆ (𝐴𝐴))
2619, 25eqsstr3d 3800 . . . . . 6 (𝐴 ∈ TosetRel → 𝐴 ⊆ (𝐴𝐴))
27 relcnv 5685 . . . . . . . 8 Rel 𝐴
28 relcoi1 5850 . . . . . . . 8 (Rel 𝐴 → (𝐴 ∘ ( I ↾ 𝐴)) = 𝐴)
2927, 28ax-mp 5 . . . . . . 7 (𝐴 ∘ ( I ↾ 𝐴)) = 𝐴
30 relcnvfld 5852 . . . . . . . . . . 11 (Rel 𝐴 𝐴 = 𝐴)
313, 30syl 17 . . . . . . . . . 10 (𝐴 ∈ TosetRel → 𝐴 = 𝐴)
3231reseq2d 5565 . . . . . . . . 9 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) = ( I ↾ 𝐴))
3332, 7eqsstr3d 3800 . . . . . . . 8 (𝐴 ∈ TosetRel → ( I ↾ 𝐴) ⊆ 𝐴)
34 coss2 5447 . . . . . . . 8 (( I ↾ 𝐴) ⊆ 𝐴 → (𝐴 ∘ ( I ↾ 𝐴)) ⊆ (𝐴𝐴))
3533, 34syl 17 . . . . . . 7 (𝐴 ∈ TosetRel → (𝐴 ∘ ( I ↾ 𝐴)) ⊆ (𝐴𝐴))
3629, 35syl5eqssr 3810 . . . . . 6 (𝐴 ∈ TosetRel → 𝐴 ⊆ (𝐴𝐴))
3726, 36unssd 3951 . . . . 5 (𝐴 ∈ TosetRel → (𝐴𝐴) ⊆ (𝐴𝐴))
3817, 37sstrd 3771 . . . 4 (𝐴 ∈ TosetRel → (dom 𝐴 × dom 𝐴) ⊆ (𝐴𝐴))
3914, 38eqsstr3d 3800 . . 3 (𝐴 ∈ TosetRel → ( 𝐴 × 𝐴) ⊆ (𝐴𝐴))
4010, 39jca 507 . 2 (𝐴 ∈ TosetRel → ((𝐴𝐴) ⊆ 𝐴 ∧ ( 𝐴 × 𝐴) ⊆ (𝐴𝐴)))
41 eqid 2765 . . 3 𝐴 = 𝐴
4241isdir 17498 . 2 (𝐴 ∈ TosetRel → (𝐴 ∈ DirRel ↔ ((Rel 𝐴 ∧ ( I ↾ 𝐴) ⊆ 𝐴) ∧ ((𝐴𝐴) ⊆ 𝐴 ∧ ( 𝐴 × 𝐴) ⊆ (𝐴𝐴)))))
438, 40, 42mpbir2and 704 1 (𝐴 ∈ TosetRel → 𝐴 ∈ DirRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  cun 3730  cin 3731  wss 3732   cuni 4594   I cid 5184   × cxp 5275  ccnv 5276  dom cdm 5277  ran crn 5278  cres 5279  ccom 5281  Rel wrel 5282  PosetRelcps 17464   TosetRel ctsr 17465  DirRelcdir 17494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-fun 6070  df-ps 17466  df-tsr 17467  df-dir 17496
This theorem is referenced by: (None)
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