MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fullsubc Structured version   Visualization version   GIF version

Theorem fullsubc 17736
Description: The full subcategory generated by a subset of objects is the category with these objects and the same morphisms as the original. The result is always a subcategory (and it is full, meaning that all morphisms of the original category between objects in the subcategory is also in the subcategory), see definition 4.1(2) of [Adamek] p. 48. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
fullsubc.b 𝐵 = (Base‘𝐶)
fullsubc.h 𝐻 = (Homf𝐶)
fullsubc.c (𝜑𝐶 ∈ Cat)
fullsubc.s (𝜑𝑆𝐵)
Assertion
Ref Expression
fullsubc (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐶))

Proof of Theorem fullsubc
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fullsubc.h . . . . 5 𝐻 = (Homf𝐶)
2 fullsubc.b . . . . 5 𝐵 = (Base‘𝐶)
31, 2homffn 17573 . . . 4 𝐻 Fn (𝐵 × 𝐵)
42fvexi 6856 . . . 4 𝐵 ∈ V
5 sscres 17706 . . . 4 ((𝐻 Fn (𝐵 × 𝐵) ∧ 𝐵 ∈ V) → (𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻)
63, 4, 5mp2an 690 . . 3 (𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻
76a1i 11 . 2 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻)
8 eqid 2736 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
9 eqid 2736 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
10 fullsubc.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
1110adantr 481 . . . . . 6 ((𝜑𝑥𝑆) → 𝐶 ∈ Cat)
12 fullsubc.s . . . . . . 7 (𝜑𝑆𝐵)
1312sselda 3944 . . . . . 6 ((𝜑𝑥𝑆) → 𝑥𝐵)
142, 8, 9, 11, 13catidcl 17562 . . . . 5 ((𝜑𝑥𝑆) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
15 simpr 485 . . . . . . 7 ((𝜑𝑥𝑆) → 𝑥𝑆)
1615, 15ovresd 7521 . . . . . 6 ((𝜑𝑥𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) = (𝑥𝐻𝑥))
171, 2, 8, 13, 13homfval 17572 . . . . . 6 ((𝜑𝑥𝑆) → (𝑥𝐻𝑥) = (𝑥(Hom ‘𝐶)𝑥))
1816, 17eqtrd 2776 . . . . 5 ((𝜑𝑥𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) = (𝑥(Hom ‘𝐶)𝑥))
1914, 18eleqtrrd 2841 . . . 4 ((𝜑𝑥𝑆) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥))
20 eqid 2736 . . . . . . . . . 10 (comp‘𝐶) = (comp‘𝐶)
2111ad3antrrr 728 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐶 ∈ Cat)
2213ad3antrrr 728 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥𝐵)
2312adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑆𝐵)
2423sselda 3944 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑦𝐵)
2524adantr 481 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑦𝐵)
2625adantr 481 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦𝐵)
2723adantr 481 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑆𝐵)
2827sselda 3944 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑧𝐵)
2928adantr 481 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
30 simprl 769 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
31 simprr 771 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
322, 8, 20, 21, 22, 26, 29, 30, 31catcocl 17565 . . . . . . . . 9 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
3315ad3antrrr 728 . . . . . . . . . . 11 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥𝑆)
34 simplr 767 . . . . . . . . . . 11 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝑆)
3533, 34ovresd 7521 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑥𝐻𝑧))
361, 2, 8, 22, 29homfval 17572 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥𝐻𝑧) = (𝑥(Hom ‘𝐶)𝑧))
3735, 36eqtrd 2776 . . . . . . . . 9 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑥(Hom ‘𝐶)𝑧))
3832, 37eleqtrrd 2841 . . . . . . . 8 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧))
3938ralrimivva 3197 . . . . . . 7 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧))
40 simplr 767 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑥𝑆)
41 simpr 485 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑦𝑆)
4240, 41ovresd 7521 . . . . . . . . . 10 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥𝐻𝑦))
4313adantr 481 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑥𝐵)
441, 2, 8, 43, 24homfval 17572 . . . . . . . . . 10 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
4542, 44eqtrd 2776 . . . . . . . . 9 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐶)𝑦))
4645adantr 481 . . . . . . . 8 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐶)𝑦))
47 simplr 767 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑦𝑆)
48 simpr 485 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑧𝑆)
4947, 48ovresd 7521 . . . . . . . . . 10 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑦𝐻𝑧))
501, 2, 8, 25, 28homfval 17572 . . . . . . . . . 10 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑦𝐻𝑧) = (𝑦(Hom ‘𝐶)𝑧))
5149, 50eqtrd 2776 . . . . . . . . 9 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑦(Hom ‘𝐶)𝑧))
5251raleqdv 3313 . . . . . . . 8 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧) ↔ ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧)))
5346, 52raleqbidv 3319 . . . . . . 7 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (∀𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧) ↔ ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧)))
5439, 53mpbird 256 . . . . . 6 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → ∀𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧))
5554ralrimiva 3143 . . . . 5 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → ∀𝑧𝑆𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧))
5655ralrimiva 3143 . . . 4 ((𝜑𝑥𝑆) → ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧))
5719, 56jca 512 . . 3 ((𝜑𝑥𝑆) → (((Id‘𝐶)‘𝑥) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧)))
5857ralrimiva 3143 . 2 (𝜑 → ∀𝑥𝑆 (((Id‘𝐶)‘𝑥) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧)))
59 xpss12 5648 . . . . 5 ((𝑆𝐵𝑆𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
6012, 12, 59syl2anc 584 . . . 4 (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
61 fnssres 6624 . . . 4 ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
623, 60, 61sylancr 587 . . 3 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
631, 9, 20, 10, 62issubc2 17722 . 2 (𝜑 → ((𝐻 ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐶) ↔ ((𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻 ∧ ∀𝑥𝑆 (((Id‘𝐶)‘𝑥) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦)∀𝑔 ∈ (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧)))))
647, 58, 63mpbir2and 711 1 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  wss 3910  cop 4592   class class class wbr 5105   × cxp 5631  cres 5635   Fn wfn 6491  cfv 6496  (class class class)co 7357  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Idccid 17545  Homf chomf 17546  cat cssc 17690  Subcatcsubc 17692
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-pm 8768  df-ixp 8836  df-cat 17548  df-cid 17549  df-homf 17550  df-ssc 17693  df-subc 17695
This theorem is referenced by:  resscat  17738  funcres2c  17788  ressffth  17825  funcsetcres2  17979
  Copyright terms: Public domain W3C validator