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

Theorem fullsubc 17900
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 17737 . . . 4 𝐻 Fn (𝐵 × 𝐵)
42fvexi 6920 . . . 4 𝐵 ∈ V
5 sscres 17870 . . . 4 ((𝐻 Fn (𝐵 × 𝐵) ∧ 𝐵 ∈ V) → (𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻)
63, 4, 5mp2an 692 . . 3 (𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻
76a1i 11 . 2 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻)
8 eqid 2734 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
9 eqid 2734 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
10 fullsubc.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
1110adantr 480 . . . . . 6 ((𝜑𝑥𝑆) → 𝐶 ∈ Cat)
12 fullsubc.s . . . . . . 7 (𝜑𝑆𝐵)
1312sselda 3994 . . . . . 6 ((𝜑𝑥𝑆) → 𝑥𝐵)
142, 8, 9, 11, 13catidcl 17726 . . . . 5 ((𝜑𝑥𝑆) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
15 simpr 484 . . . . . . 7 ((𝜑𝑥𝑆) → 𝑥𝑆)
1615, 15ovresd 7599 . . . . . 6 ((𝜑𝑥𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) = (𝑥𝐻𝑥))
171, 2, 8, 13, 13homfval 17736 . . . . . 6 ((𝜑𝑥𝑆) → (𝑥𝐻𝑥) = (𝑥(Hom ‘𝐶)𝑥))
1816, 17eqtrd 2774 . . . . 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 2734 . . . . . . . . . 10 (comp‘𝐶) = (comp‘𝐶)
2111ad3antrrr 730 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐶 ∈ Cat)
2213ad3antrrr 730 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥𝐵)
2312adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑆𝐵)
2423sselda 3994 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑦𝐵)
2524adantr 480 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑦𝐵)
2625adantr 480 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦𝐵)
2723adantr 480 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑆𝐵)
2827sselda 3994 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑧𝐵)
2928adantr 480 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
30 simprl 771 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
31 simprr 773 . . . . . . . . . 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 17729 . . . . . . . . 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 730 . . . . . . . . . . 11 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥𝑆)
34 simplr 769 . . . . . . . . . . 11 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝑆)
3533, 34ovresd 7599 . . . . . . . . . 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 17736 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥𝐻𝑧) = (𝑥(Hom ‘𝐶)𝑧))
3735, 36eqtrd 2774 . . . . . . . . 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 3199 . . . . . . 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 769 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑥𝑆)
41 simpr 484 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑦𝑆)
4240, 41ovresd 7599 . . . . . . . . . 10 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥𝐻𝑦))
4313adantr 480 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑥𝐵)
441, 2, 8, 43, 24homfval 17736 . . . . . . . . . 10 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
4542, 44eqtrd 2774 . . . . . . . . 9 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐶)𝑦))
4645adantr 480 . . . . . . . 8 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐶)𝑦))
47 simplr 769 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑦𝑆)
48 simpr 484 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑧𝑆)
4947, 48ovresd 7599 . . . . . . . . . 10 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑦𝐻𝑧))
501, 2, 8, 25, 28homfval 17736 . . . . . . . . . 10 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑦𝐻𝑧) = (𝑦(Hom ‘𝐶)𝑧))
5149, 50eqtrd 2774 . . . . . . . . 9 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑦(Hom ‘𝐶)𝑧))
5251raleqdv 3323 . . . . . . . 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 3343 . . . . . . 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 257 . . . . . 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 511 . . 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 5703 . . . . 5 ((𝑆𝐵𝑆𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
6012, 12, 59syl2anc 584 . . . 4 (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
61 fnssres 6691 . . . 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 17886 . 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 713 1 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  wss 3962  cop 4636   class class class wbr 5147   × cxp 5686  cres 5690   Fn wfn 6557  cfv 6562  (class class class)co 7430  Basecbs 17244  Hom chom 17308  compcco 17309  Catccat 17708  Idccid 17709  Homf chomf 17710  cat cssc 17854  Subcatcsubc 17856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-pm 8867  df-ixp 8936  df-cat 17712  df-cid 17713  df-homf 17714  df-ssc 17857  df-subc 17859
This theorem is referenced by:  resscat  17902  funcres2c  17954  ressffth  17991  funcsetcres2  18146
  Copyright terms: Public domain W3C validator