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Theorem resspos 32941
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 7466 . 2 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (𝐹s 𝐴) ∈ V)
2 eqid 2735 . . . . . . 7 (𝐹s 𝐴) = (𝐹s 𝐴)
3 eqid 2735 . . . . . . 7 (Base‘𝐹) = (Base‘𝐹)
42, 3ressbas 17280 . . . . . 6 (𝐴𝑉 → (𝐴 ∩ (Base‘𝐹)) = (Base‘(𝐹s 𝐴)))
5 inss2 4246 . . . . . 6 (𝐴 ∩ (Base‘𝐹)) ⊆ (Base‘𝐹)
64, 5eqsstrrdi 4051 . . . . 5 (𝐴𝑉 → (Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹))
76adantl 481 . . . 4 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (Base‘(𝐹s 𝐴)) ⊆ (Base‘𝐹))
8 eqid 2735 . . . . . . 7 (le‘𝐹) = (le‘𝐹)
93, 8ispos 18372 . . . . . 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 4064 . . . . . . . 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 3167 . . . . . . 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 4064 . . . . . . 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 3167 . . . . 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 4064 . . . . 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 17426 . . . . 5 (𝐴𝑉 → (le‘𝐹) = (le‘(𝐹s 𝐴)))
2120adantl 481 . . . 4 ((𝐹 ∈ Poset ∧ 𝐴𝑉) → (le‘𝐹) = (le‘(𝐹s 𝐴)))
22 breq 5150 . . . . . . 7 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑥𝑥(le‘(𝐹s 𝐴))𝑥))
23 breq 5150 . . . . . . . . 9 ((le‘𝐹) = (le‘(𝐹s 𝐴)) → (𝑥(le‘𝐹)𝑦𝑥(le‘(𝐹s 𝐴))𝑦))
24 breq 5150 . . . . . . . . 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 5150 . . . . . . . . 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 5150 . . . . . . . 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 1435 . . . . . 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 3176 . . . . 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 3219 . . . 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 2735 . . 3 (Base‘(𝐹s 𝐴)) = (Base‘(𝐹s 𝐴))
37 eqid 2735 . . 3 (le‘(𝐹s 𝐴)) = (le‘(𝐹s 𝐴))
3836, 37ispos 18372 . 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 1537  wcel 2106  wral 3059  Vcvv 3478  cin 3962  wss 3963   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  lecple 17305  Posetcpo 18365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-dec 12732  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-ple 17318  df-poset 18371
This theorem is referenced by:  resstos  32942
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