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Theorem tsrss 17144
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 17135 . . 3 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
2 inss1 3811 . . . . . 6 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅
3 dmss 5283 . . . . . 6 ((𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝑅 → dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ dom 𝑅)
4 ssralv 3645 . . . . . 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 3645 . . . . . . 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 2947 . . . . 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 3812 . . . . . . . . . 10 (𝑅 ∩ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
11 dmss 5283 . . . . . . . . . 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 5305 . . . . . . . . 9 dom (𝐴 × 𝐴) = 𝐴
1412, 13sseqtri 3616 . . . . . . . 8 dom (𝑅 ∩ (𝐴 × 𝐴)) ⊆ 𝐴
1514sseli 3579 . . . . . . 7 (𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑥𝐴)
1614sseli 3579 . . . . . . 7 (𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) → 𝑦𝐴)
17 brinxp 5142 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
18 brinxp 5142 . . . . . . . . 9 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
1918ancoms 469 . . . . . . . 8 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2017, 19orbi12d 745 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
2115, 16, 20syl2an 494 . . . . . 6 ((𝑥 ∈ dom (𝑅 ∩ (𝐴 × 𝐴)) ∧ 𝑦 ∈ dom (𝑅 ∩ (𝐴 × 𝐴))) → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
2221ralbidva 2979 . . . . 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 2973 . . . 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 208 . . 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 589 . 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 2621 . . 3 dom 𝑅 = dom 𝑅
2726istsr2 17139 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅(𝑥𝑅𝑦𝑦𝑅𝑥)))
28 eqid 2621 . . 3 dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
2928istsr2 17139 . 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 281 1 (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ TosetRel )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  wcel 1987  wral 2907  cin 3554  wss 3555   class class class wbr 4613   × cxp 5072  dom cdm 5074  PosetRelcps 17119   TosetRel ctsr 17120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ps 17121  df-tsr 17122
This theorem is referenced by: (None)
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