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Theorem resspos 32957
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 7467 . 2 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ V)
2 eqid 2736 . . . . . . 7 (𝐹s 𝐴) = (𝐹s 𝐴)
3 eqid 2736 . . . . . . 7 (Base‘𝐹) = (Base‘𝐹)
42, 3ressbas 17281 . . . . . 6 (𝐴𝑉 → (𝐴 ∩ (Base‘𝐹)) = (Base‘(𝐹s 𝐴)))
5 inss2 4237 . . . . . 6 (𝐴 ∩ (Base‘𝐹)) ⊆ (Base‘𝐹)
64, 5eqsstrrdi 4028 . . . . 5 (𝐴𝑉 → (Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹))
76adantl 481 . . . 4 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹))
8 eqid 2736 . . . . . . 7 (le‘𝐹) = (le‘𝐹)
93, 8ispos 18361 . . . . . 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 496 . . . . 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 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)))
12 ssralv 4051 . . . . . . . 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 3168 . . . . . . 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 4051 . . . . . . 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 3168 . . . . 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 4051 . . . . 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 17425 . . . . 5 (𝐴𝑉 → (le‘𝐹) = (le‘(𝐹s 𝐴)))
2120adantl 481 . . . 4 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (le‘𝐹) = (le‘(𝐹s 𝐴)))
22 breq 5144 . . . . . . 7 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑥𝑥(le‘(𝐹s 𝐴))𝑥))
23 breq 5144 . . . . . . . . 9 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑦𝑥(le‘(𝐹s 𝐴))𝑦))
24 breq 5144 . . . . . . . . 9 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑦(le‘𝐹)𝑥𝑦(le‘(𝐹s 𝐴))𝑥))
2523, 24anbi12d 632 . . . . . . . 8 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥) ↔ (𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑥)))
2625imbi1d 341 . . . . . . 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 5144 . . . . . . . . 9 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑦(le‘𝐹)𝑧𝑦(le‘(𝐹s 𝐴))𝑧))
2823, 27anbi12d 632 . . . . . . . 8 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → ((𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑧) ↔ (𝑥(le‘(𝐹s 𝐴))𝑦𝑦(le‘(𝐹s 𝐴))𝑧)))
29 breq 5144 . . . . . . . 8 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑧𝑥(le‘(𝐹s 𝐴))𝑧))
3028, 29imbi12d 344 . . . . . . 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 1437 . . . . . 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 3177 . . . . 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 3220 . . . 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 232 . 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 2736 . . 3 (Base‘(𝐹s 𝐴)) = (Base‘(𝐹s 𝐴))
37 eqid 2736 . . 3 (le‘(𝐹s 𝐴)) = (le‘(𝐹s 𝐴))
3836, 37ispos 18361 . 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 583 1 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  cin 3949  wss 3950   class class class wbr 5142  cfv 6560  (class class class)co 7432  Basecbs 17248  s cress 17275  lecple 17305  Posetcpo 18354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-dec 12736  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-ple 17318  df-poset 18360
This theorem is referenced by:  resstos  32958
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