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Theorem istos 16858
 Description: The predicate "is a toset." (Contributed by FL, 17-Nov-2014.)
Hypotheses
Ref Expression
istos.b 𝐵 = (Base‘𝐾)
istos.l = (le‘𝐾)
Assertion
Ref Expression
istos (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥, ,𝑦
Allowed substitution hints:   𝐾(𝑥,𝑦)

Proof of Theorem istos
Dummy variables 𝑓 𝑏 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . . 4 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
2 fveq2 6103 . . . . 5 (𝑓 = 𝐾 → (le‘𝑓) = (le‘𝐾))
32sbceq1d 3407 . . . 4 (𝑓 = 𝐾 → ([(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ [(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥)))
41, 3sbceqbid 3409 . . 3 (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ [(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥)))
5 fvex 6113 . . . 4 (Base‘𝐾) ∈ V
6 fvex 6113 . . . 4 (le‘𝐾) ∈ V
7 istos.b . . . . . 6 𝐵 = (Base‘𝐾)
8 eqtr 2629 . . . . . . . . 9 ((𝑏 = (Base‘𝐾) ∧ (Base‘𝐾) = 𝐵) → 𝑏 = 𝐵)
9 istos.l . . . . . . . . . 10 = (le‘𝐾)
10 eqtr 2629 . . . . . . . . . . . . 13 ((𝑟 = (le‘𝐾) ∧ (le‘𝐾) = ) → 𝑟 = )
11 breq 4585 . . . . . . . . . . . . . . . . 17 (𝑟 = → (𝑥𝑟𝑦𝑥 𝑦))
12 breq 4585 . . . . . . . . . . . . . . . . 17 (𝑟 = → (𝑦𝑟𝑥𝑦 𝑥))
1311, 12orbi12d 742 . . . . . . . . . . . . . . . 16 (𝑟 = → ((𝑥𝑟𝑦𝑦𝑟𝑥) ↔ (𝑥 𝑦𝑦 𝑥)))
14132ralbidv 2972 . . . . . . . . . . . . . . 15 (𝑟 = → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝑏𝑦𝑏 (𝑥 𝑦𝑦 𝑥)))
15 raleq 3115 . . . . . . . . . . . . . . . 16 (𝑏 = 𝐵 → (∀𝑦𝑏 (𝑥 𝑦𝑦 𝑥) ↔ ∀𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
1615raleqbi1dv 3123 . . . . . . . . . . . . . . 15 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥 𝑦𝑦 𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
1714, 16sylan9bb 732 . . . . . . . . . . . . . 14 ((𝑟 = 𝑏 = 𝐵) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
1817ex 449 . . . . . . . . . . . . 13 (𝑟 = → (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))))
1910, 18syl 17 . . . . . . . . . . . 12 ((𝑟 = (le‘𝐾) ∧ (le‘𝐾) = ) → (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))))
2019expcom 450 . . . . . . . . . . 11 ((le‘𝐾) = → (𝑟 = (le‘𝐾) → (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))))
2120eqcoms 2618 . . . . . . . . . 10 ( = (le‘𝐾) → (𝑟 = (le‘𝐾) → (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))))
229, 21ax-mp 5 . . . . . . . . 9 (𝑟 = (le‘𝐾) → (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))))
238, 22syl5com 31 . . . . . . . 8 ((𝑏 = (Base‘𝐾) ∧ (Base‘𝐾) = 𝐵) → (𝑟 = (le‘𝐾) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))))
2423expcom 450 . . . . . . 7 ((Base‘𝐾) = 𝐵 → (𝑏 = (Base‘𝐾) → (𝑟 = (le‘𝐾) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))))
2524eqcoms 2618 . . . . . 6 (𝐵 = (Base‘𝐾) → (𝑏 = (Base‘𝐾) → (𝑟 = (le‘𝐾) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))))
267, 25ax-mp 5 . . . . 5 (𝑏 = (Base‘𝐾) → (𝑟 = (le‘𝐾) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))))
2726imp 444 . . . 4 ((𝑏 = (Base‘𝐾) ∧ 𝑟 = (le‘𝐾)) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
285, 6, 27sbc2ie 3472 . . 3 ([(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))
294, 28syl6bb 275 . 2 (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
30 df-toset 16857 . 2 Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥)}
3129, 30elrab2 3333 1 (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  [wsbc 3402   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763  Tosetctos 16856 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-toset 16857 This theorem is referenced by:  tosso  16859  zntoslem  19724  tospos  28989  resstos  28991  tleile  28992  odutos  28994  xrstos  29010  xrge0omnd  29042
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