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Theorem resspos 28990
 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 ovex 6577 . . 3 (𝐹s 𝐴) ∈ V
21a1i 11 . 2 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ V)
3 eqid 2610 . . . . . . 7 (𝐹s 𝐴) = (𝐹s 𝐴)
4 eqid 2610 . . . . . . 7 (Base‘𝐹) = (Base‘𝐹)
53, 4ressbas 15757 . . . . . 6 (𝐴𝑉 → (𝐴 ∩ (Base‘𝐹)) = (Base‘(𝐹s 𝐴)))
6 inss2 3796 . . . . . 6 (𝐴 ∩ (Base‘𝐹)) ⊆ (Base‘𝐹)
75, 6syl6eqssr 3619 . . . . 5 (𝐴𝑉 → (Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹))
87adantl 481 . . . 4 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹))
9 eqid 2610 . . . . . . 7 (le‘𝐹) = (le‘𝐹)
104, 9ispos 16770 . . . . . 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))))
1110simprbi 479 . . . . 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)))
1211adantr 480 . . . 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)))
13 ssralv 3629 . . . . . . . 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))))
1413ralimdv 2946 . . . . . . 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))))
15 ssralv 3629 . . . . . . 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))))
1614, 15syld 46 . . . . . 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))))
1716ralimdv 2946 . . . . 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))))
18 ssralv 3629 . . . . 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))))
1917, 18syld 46 . . . 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))))
208, 12, 19sylc 63 . . 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)))
213, 9ressle 15882 . . . . 5 (𝐴𝑉 → (le‘𝐹) = (le‘(𝐹s 𝐴)))
2221adantl 481 . . . 4 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (le‘𝐹) = (le‘(𝐹s 𝐴)))
23 breq 4585 . . . . . . 7 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑥𝑥(le‘(𝐹s 𝐴))𝑥))
24 breq 4585 . . . . . . . . 9 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑦𝑥(le‘(𝐹s 𝐴))𝑦))
25 breq 4585 . . . . . . . . 9 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑦(le‘𝐹)𝑥𝑦(le‘(𝐹s 𝐴))𝑥))
2624, 25anbi12d 743 . . . . . . . 8 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) ↔ (𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥)))
2726imbi1d 330 . . . . . . 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)))
28 breq 4585 . . . . . . . . 9 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑦(le‘𝐹)𝑧𝑦(le‘(𝐹s 𝐴))𝑧))
2924, 28anbi12d 743 . . . . . . . 8 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) ↔ (𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧)))
30 breq 4585 . . . . . . . 8 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑧𝑥(le‘(𝐹s 𝐴))𝑧))
3129, 30imbi12d 333 . . . . . . 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)))
3223, 27, 313anbi123d 1391 . . . . . 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))))
3332ralbidv 2969 . . . . 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))))
34332ralbidv 2972 . . . 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))))
3522, 34syl 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))))
3620, 35mpbid 221 . 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)))
37 eqid 2610 . . 3 (Base‘(𝐹s 𝐴)) = (Base‘(𝐹s 𝐴))
38 eqid 2610 . . 3 (le‘(𝐹s 𝐴)) = (le‘(𝐹s 𝐴))
3937, 38ispos 16770 . 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))))
402, 36, 39sylanbrc 695 1 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Poset)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  lecple 15775  Posetcpo 16763 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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-dec 11370  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-ple 15788  df-poset 16769 This theorem is referenced by:  resstos  28991
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