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Theorem tsrss 17691
Description: Any subset of a totally ordered set is totally ordered. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Nov-2013.)
Assertion
Ref Expression
tsrss (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ TosetRel )

Proof of Theorem tsrss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psss 17682 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
2 inss1 4093 . . . . . 6 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
3 dmss 5621 . . . . . 6 ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom 𝑅)
4 ssralv 3924 . . . . . 6 (dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom 𝑅 → (∀𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅(𝑥𝑅𝑦𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦𝑦𝑅𝑥)))
52, 3, 4mp2b 10 . . . . 5 (∀𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅(𝑥𝑅𝑦𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦𝑦𝑅𝑥))
6 ssralv 3924 . . . . . . 7 (dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom 𝑅 → (∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦𝑦𝑅𝑥) → ∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦𝑦𝑅𝑥)))
72, 3, 6mp2b 10 . . . . . 6 (∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦𝑦𝑅𝑥) → ∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦𝑦𝑅𝑥))
87ralimi 3111 . . . . 5 (∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom 𝑅(𝑥𝑅𝑦𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦𝑦𝑅𝑥))
95, 8syl 17 . . . 4 (∀𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅(𝑥𝑅𝑦𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦𝑦𝑅𝑥))
10 inss2 4094 . . . . . . . . . 10 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
11 dmss 5621 . . . . . . . . . 10 ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom (𝐴 × 𝐴))
1210, 11ax-mp 5 . . . . . . . . 9 dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom (𝐴 × 𝐴)
13 dmxpid 5643 . . . . . . . . 9 dom (𝐴 × 𝐴) = 𝐴
1412, 13sseqtri 3894 . . . . . . . 8 dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝐴
1514sseli 3855 . . . . . . 7 (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑥𝐴)
1614sseli 3855 . . . . . . 7 (𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑦𝐴)
17 brinxp 5480 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
18 brinxp 5480 . . . . . . . . 9 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1918ancoms 451 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2017, 19orbi12d 902 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
2115, 16, 20syl2an 586 . . . . . 6 ((𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∧ 𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))) → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
2221ralbidva 3147 . . . . 5 (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → (∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
2322ralbiia 3115 . . . 4 (∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥𝑅𝑦𝑦𝑅𝑥) ↔ ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
249, 23sylib 210 . . 3 (∀𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅(𝑥𝑅𝑦𝑦𝑅𝑥) → ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
251, 24anim12i 603 . 2 ((𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅(𝑥𝑅𝑦𝑦𝑅𝑥)) → ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel ∧ ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
26 eqid 2779 . . 3 dom 𝑅 = dom 𝑅
2726istsr2 17686 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅(𝑥𝑅𝑦𝑦𝑅𝑥)))
28 eqid 2779 . . 3 dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
2928istsr2 17686 . 2 ((𝑅 ∩ (𝐴 × 𝐴)) ∈ TosetRel ↔ ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel ∧ ∀𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))∀𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))(𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
3025, 27, 293imtr4i 284 1 (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ TosetRel )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wo 833  wcel 2050  wral 3089  cin 3829  wss 3830   class class class wbr 4929   × cxp 5405  dom cdm 5407  PosetRelcps 17666   TosetRel ctsr 17667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ps 17668  df-tsr 17669
This theorem is referenced by: (None)
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