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Theorem resspos 30676
Description: The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Assertion
Ref Expression
resspos ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Poset)

Proof of Theorem resspos
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovexd 7174 . 2 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ V)
2 eqid 2801 . . . . . . 7 (𝐹s 𝐴) = (𝐹s 𝐴)
3 eqid 2801 . . . . . . 7 (Base‘𝐹) = (Base‘𝐹)
42, 3ressbas 16550 . . . . . 6 (𝐴𝑉 → (𝐴 ∩ (Base‘𝐹)) = (Base‘(𝐹s 𝐴)))
5 inss2 4159 . . . . . 6 (𝐴 ∩ (Base‘𝐹)) ⊆ (Base‘𝐹)
64, 5eqsstrrdi 3973 . . . . 5 (𝐴𝑉 → (Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹))
76adantl 485 . . . 4 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹))
8 eqid 2801 . . . . . . 7 (le‘𝐹) = (le‘𝐹)
93, 8ispos 17553 . . . . . 6 (𝐹 ∈ Poset ↔ (𝐹 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
109simprbi 500 . . . . 5 (𝐹 ∈ Poset → ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)))
1110adantr 484 . . . 4 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)))
12 ssralv 3984 . . . . . . . 8 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
1312ralimdv 3148 . . . . . . 7 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
14 ssralv 3984 . . . . . . 7 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
1513, 14syld 47 . . . . . 6 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
1615ralimdv 3148 . . . . 5 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
17 ssralv 3984 . . . . 5 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
1816, 17syld 47 . . . 4 ((Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹) → (∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) → ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧))))
197, 11, 18sylc 65 . . 3 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)))
202, 8ressle 16668 . . . . 5 (𝐴𝑉 → (le‘𝐹) = (le‘(𝐹s 𝐴)))
2120adantl 485 . . . 4 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (le‘𝐹) = (le‘(𝐹s 𝐴)))
22 breq 5035 . . . . . . 7 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑥𝑥(le‘(𝐹s 𝐴))𝑥))
23 breq 5035 . . . . . . . . 9 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑦𝑥(le‘(𝐹s 𝐴))𝑦))
24 breq 5035 . . . . . . . . 9 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑦(le‘𝐹)𝑥𝑦(le‘(𝐹s 𝐴))𝑥))
2523, 24anbi12d 633 . . . . . . . 8 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) ↔ (𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥)))
2625imbi1d 345 . . . . . . 7 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ↔ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦)))
27 breq 5035 . . . . . . . . 9 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑦(le‘𝐹)𝑧𝑦(le‘(𝐹s 𝐴))𝑧))
2823, 27anbi12d 633 . . . . . . . 8 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) ↔ (𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧)))
29 breq 5035 . . . . . . . 8 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑧𝑥(le‘(𝐹s 𝐴))𝑧))
3028, 29imbi12d 348 . . . . . . 7 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧) ↔ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧)))
3122, 26, 303anbi123d 1433 . . . . . 6 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → ((𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) ↔ (𝑥(le‘(𝐹s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧))))
3231ralbidv 3165 . . . . 5 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) ↔ ∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘(𝐹s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧))))
33322ralbidv 3167 . . . 4 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) ↔ ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘(𝐹s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧))))
3421, 33syl 17 . . 3 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘𝐹)𝑥 ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) → 𝑥(le‘𝐹)𝑧)) ↔ ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘(𝐹s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧))))
3519, 34mpbid 235 . 2 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘(𝐹s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧)))
36 eqid 2801 . . 3 (Base‘(𝐹s 𝐴)) = (Base‘(𝐹s 𝐴))
37 eqid 2801 . . 3 (le‘(𝐹s 𝐴)) = (le‘(𝐹s 𝐴))
3836, 37ispos 17553 . 2 ((𝐹s 𝐴) ∈ Poset ↔ ((𝐹s 𝐴) ∈ V ∧ ∀𝑥 ∈ (Base‘(𝐹s 𝐴))∀𝑦 ∈ (Base‘(𝐹s 𝐴))∀𝑧 ∈ (Base‘(𝐹s 𝐴))(𝑥(le‘(𝐹s 𝐴))𝑥 ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧) → 𝑥(le‘(𝐹s 𝐴))𝑧))))
391, 35, 38sylanbrc 586 1 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  Vcvv 3444  cin 3883  wss 3884   class class class wbr 5033  cfv 6328  (class class class)co 7139  Basecbs 16479  s cress 16480  lecple 16568  Posetcpo 17546
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-dec 12091  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-ple 16581  df-poset 17552
This theorem is referenced by:  resstos  30677
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