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Theorem isprs 17245
Description: Property of being a preordered set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypotheses
Ref Expression
isprs.b 𝐵 = (Base‘𝐾)
isprs.l = (le‘𝐾)
Assertion
Ref Expression
isprs (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
Distinct variable groups:   𝑥,𝐾,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥, ,𝑦,𝑧

Proof of Theorem isprs
Dummy variables 𝑓 𝑏 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6411 . . . 4 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
2 fveq2 6411 . . . . 5 (𝑓 = 𝐾 → (le‘𝑓) = (le‘𝐾))
32sbceq1d 3638 . . . 4 (𝑓 = 𝐾 → ([(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ [(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
41, 3sbceqbid 3640 . . 3 (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ [(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
5 fvex 6424 . . . 4 (Base‘𝐾) ∈ V
6 fvex 6424 . . . 4 (le‘𝐾) ∈ V
7 isprs.b . . . . . . 7 𝐵 = (Base‘𝐾)
8 eqtr3 2820 . . . . . . 7 ((𝑏 = (Base‘𝐾) ∧ 𝐵 = (Base‘𝐾)) → 𝑏 = 𝐵)
97, 8mpan2 683 . . . . . 6 (𝑏 = (Base‘𝐾) → 𝑏 = 𝐵)
10 raleq 3321 . . . . . . . 8 (𝑏 = 𝐵 → (∀𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
1110raleqbi1dv 3329 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
1211raleqbi1dv 3329 . . . . . 6 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
139, 12syl 17 . . . . 5 (𝑏 = (Base‘𝐾) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
14 isprs.l . . . . . . 7 = (le‘𝐾)
15 eqtr3 2820 . . . . . . 7 ((𝑟 = (le‘𝐾) ∧ = (le‘𝐾)) → 𝑟 = )
1614, 15mpan2 683 . . . . . 6 (𝑟 = (le‘𝐾) → 𝑟 = )
17 breq 4845 . . . . . . . . 9 (𝑟 = → (𝑥𝑟𝑥𝑥 𝑥))
18 breq 4845 . . . . . . . . . . 11 (𝑟 = → (𝑥𝑟𝑦𝑥 𝑦))
19 breq 4845 . . . . . . . . . . 11 (𝑟 = → (𝑦𝑟𝑧𝑦 𝑧))
2018, 19anbi12d 625 . . . . . . . . . 10 (𝑟 = → ((𝑥𝑟𝑦𝑦𝑟𝑧) ↔ (𝑥 𝑦𝑦 𝑧)))
21 breq 4845 . . . . . . . . . 10 (𝑟 = → (𝑥𝑟𝑧𝑥 𝑧))
2220, 21imbi12d 336 . . . . . . . . 9 (𝑟 = → (((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
2317, 22anbi12d 625 . . . . . . . 8 (𝑟 = → ((𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
2423ralbidv 3167 . . . . . . 7 (𝑟 = → (∀𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
25242ralbidv 3170 . . . . . 6 (𝑟 = → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
2616, 25syl 17 . . . . 5 (𝑟 = (le‘𝐾) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
2713, 26sylan9bb 506 . . . 4 ((𝑏 = (Base‘𝐾) ∧ 𝑟 = (le‘𝐾)) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
285, 6, 27sbc2ie 3701 . . 3 ([(Base‘𝐾) / 𝑏][(le‘𝐾) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
294, 28syl6bb 279 . 2 (𝑓 = 𝐾 → ([(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
30 df-proset 17243 . 2 Proset = {𝑓[(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
3129, 30elab4g 3547 1 (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  [wsbc 3633   class class class wbr 4843  cfv 6101  Basecbs 16184  lecple 16274   Proset cproset 17241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-nul 4983
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-proset 17243
This theorem is referenced by:  prslem  17246  ispos2  17263  ressprs  30171  oduprs  30172
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