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Theorem istos 16967
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 6153 . . . 4 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
2 fveq2 6153 . . . . 5 (𝑓 = 𝐾 → (le‘𝑓) = (le‘𝐾))
32sbceq1d 3426 . . . 4 (𝑓 = 𝐾 → ([(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ [(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥)))
41, 3sbceqbid 3428 . . 3 (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ [(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥)))
5 fvex 6163 . . . 4 (Base‘𝐾) ∈ V
6 fvex 6163 . . . 4 (le‘𝐾) ∈ V
7 istos.b . . . . . 6 𝐵 = (Base‘𝐾)
8 eqtr 2640 . . . . . . . . 9 ((𝑏 = (Base‘𝐾) ∧ (Base‘𝐾) = 𝐵) → 𝑏 = 𝐵)
9 istos.l . . . . . . . . . 10 = (le‘𝐾)
10 eqtr 2640 . . . . . . . . . . . . 13 ((𝑟 = (le‘𝐾) ∧ (le‘𝐾) = ) → 𝑟 = )
11 breq 4620 . . . . . . . . . . . . . . . . 17 (𝑟 = → (𝑥𝑟𝑦𝑥 𝑦))
12 breq 4620 . . . . . . . . . . . . . . . . 17 (𝑟 = → (𝑦𝑟𝑥𝑦 𝑥))
1311, 12orbi12d 745 . . . . . . . . . . . . . . . 16 (𝑟 = → ((𝑥𝑟𝑦𝑦𝑟𝑥) ↔ (𝑥 𝑦𝑦 𝑥)))
14132ralbidv 2984 . . . . . . . . . . . . . . 15 (𝑟 = → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝑏𝑦𝑏 (𝑥 𝑦𝑦 𝑥)))
15 raleq 3130 . . . . . . . . . . . . . . . 16 (𝑏 = 𝐵 → (∀𝑦𝑏 (𝑥 𝑦𝑦 𝑥) ↔ ∀𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
1615raleqbi1dv 3138 . . . . . . . . . . . . . . 15 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥 𝑦𝑦 𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
1714, 16sylan9bb 735 . . . . . . . . . . . . . 14 ((𝑟 = 𝑏 = 𝐵) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
1817ex 450 . . . . . . . . . . . . 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 451 . . . . . . . . . . 11 ((le‘𝐾) = → (𝑟 = (le‘𝐾) → (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))))
2120eqcoms 2629 . . . . . . . . . 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 451 . . . . . . 7 ((Base‘𝐾) = 𝐵 → (𝑏 = (Base‘𝐾) → (𝑟 = (le‘𝐾) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))))
2524eqcoms 2629 . . . . . 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 445 . . . 4 ((𝑏 = (Base‘𝐾) ∧ 𝑟 = (le‘𝐾)) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
285, 6, 27sbc2ie 3491 . . 3 ([(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥))
294, 28syl6bb 276 . 2 (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
30 df-toset 16966 . 2 Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏 (𝑥𝑟𝑦𝑦𝑟𝑥)}
3129, 30elrab2 3352 1 (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wral 2907  [wsbc 3421   class class class wbr 4618  cfv 5852  Basecbs 15792  lecple 15880  Posetcpo 16872  Tosetctos 16965
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-nul 4754
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-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-toset 16966
This theorem is referenced by:  tosso  16968  zntoslem  19837  tospos  29467  resstos  29469  tleile  29470  odutos  29472  xrstos  29488  xrge0omnd  29520
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