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Theorem istos 17624
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 6643 . . . 4 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
2 fveq2 6643 . . . . 5 (𝑓 = 𝐾 → (le‘𝑓) = (le‘𝐾))
32sbceq1d 3754 . . . 4 (𝑓 = 𝐾 → ([(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ [(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥)))
41, 3sbceqbid 3756 . . 3 (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ [(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥)))
5 fvex 6656 . . . 4 (Base‘𝐾) ∈ V
6 fvex 6656 . . . 4 (le‘𝐾) ∈ V
7 istos.b . . . . . 6 𝐵 = (Base‘𝐾)
8 eqtr 2841 . . . . . . . . 9 ((𝑏 = (Base‘𝐾) ∧ (Base‘𝐾) = 𝐵) → 𝑏 = 𝐵)
9 istos.l . . . . . . . . . 10 = (le‘𝐾)
10 eqtr 2841 . . . . . . . . . . . . 13 ((𝑟 = (le‘𝐾) ∧ (le‘𝐾) = ) → 𝑟 = )
11 breq 5041 . . . . . . . . . . . . . . . . 17 (𝑟 = → (𝑥𝑟𝑦𝑥 𝑦))
12 breq 5041 . . . . . . . . . . . . . . . . 17 (𝑟 = → (𝑦𝑟𝑥𝑦 𝑥))
1311, 12orbi12d 916 . . . . . . . . . . . . . . . 16 (𝑟 = → ((𝑥𝑟𝑦𝑦𝑟𝑥) ↔ (𝑥 𝑦𝑦 𝑥)))
14132ralbidv 3187 . . . . . . . . . . . . . . 15 (𝑟 = → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝑏𝑦𝑏 (𝑥 𝑦𝑦 𝑥)))
15 raleq 3390 . . . . . . . . . . . . . . . 16 (𝑏 = 𝐵 → (∀𝑦𝑏 (𝑥 𝑦𝑦 𝑥) ↔ ∀𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
1615raleqbi1dv 3388 . . . . . . . . . . . . . . 15 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥 𝑦𝑦 𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
1714, 16sylan9bb 513 . . . . . . . . . . . . . 14 ((𝑟 = 𝑏 = 𝐵) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
1817ex 416 . . . . . . . . . . . . 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 417 . . . . . . . . . . 11 ((le‘𝐾) = → (𝑟 = (le‘𝐾) → (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))))
2120eqcoms 2829 . . . . . . . . . 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 417 . . . . . . 7 ((Base‘𝐾) = 𝐵 → (𝑏 = (Base‘𝐾) → (𝑟 = (le‘𝐾) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))))
2524eqcoms 2829 . . . . . 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 410 . . . 4 ((𝑏 = (Base‘𝐾) ∧ 𝑟 = (le‘𝐾)) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
285, 6, 27sbc2ie 3825 . . 3 ([(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))
294, 28syl6bb 290 . 2 (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
30 df-toset 17623 . 2 Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥)}
3129, 30elrab2 3660 1 (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  wral 3126  [wsbc 3749   class class class wbr 5039  cfv 6328  Basecbs 16462  lecple 16551  Posetcpo 17529  Tosetctos 17622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-nul 5183
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336  df-toset 17623
This theorem is referenced by:  tosso  17625  zntoslem  20679  tospos  30632  resstos  30634  tleile  30635  odutos  30637  xrstos  30674  xrge0omnd  30720
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