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

Theorem fullsubc 17172
 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 17014 . . . 4 𝐻 Fn (𝐵 × 𝐵)
42fvexi 6673 . . . 4 𝐵 ∈ V
5 sscres 17145 . . . 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 2759 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
9 eqid 2759 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
10 fullsubc.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
1110adantr 485 . . . . . 6 ((𝜑𝑥𝑆) → 𝐶 ∈ Cat)
12 fullsubc.s . . . . . . 7 (𝜑𝑆𝐵)
1312sselda 3893 . . . . . 6 ((𝜑𝑥𝑆) → 𝑥𝐵)
142, 8, 9, 11, 13catidcl 17004 . . . . 5 ((𝜑𝑥𝑆) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
15 simpr 489 . . . . . . 7 ((𝜑𝑥𝑆) → 𝑥𝑆)
1615, 15ovresd 7312 . . . . . 6 ((𝜑𝑥𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) = (𝑥𝐻𝑥))
171, 2, 8, 13, 13homfval 17013 . . . . . 6 ((𝜑𝑥𝑆) → (𝑥𝐻𝑥) = (𝑥(Hom ‘𝐶)𝑥))
1816, 17eqtrd 2794 . . . . 5 ((𝜑𝑥𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) = (𝑥(Hom ‘𝐶)𝑥))
1914, 18eleqtrrd 2856 . . . 4 ((𝜑𝑥𝑆) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥))
20 eqid 2759 . . . . . . . . . 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 485 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑆𝐵)
2423sselda 3893 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑦𝐵)
2524adantr 485 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑦𝐵)
2625adantr 485 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦𝐵)
2723adantr 485 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑆𝐵)
2827sselda 3893 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑧𝐵)
2928adantr 485 . . . . . . . . . 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 17007 . . . . . . . . 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 7312 . . . . . . . . . 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 17013 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥𝐻𝑧) = (𝑥(Hom ‘𝐶)𝑧))
3735, 36eqtrd 2794 . . . . . . . . 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 2856 . . . . . . . 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 3121 . . . . . . 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 489 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑦𝑆)
4240, 41ovresd 7312 . . . . . . . . . 10 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥𝐻𝑦))
4313adantr 485 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑥𝐵)
441, 2, 8, 43, 24homfval 17013 . . . . . . . . . 10 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
4542, 44eqtrd 2794 . . . . . . . . 9 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐶)𝑦))
4645adantr 485 . . . . . . . 8 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐶)𝑦))
47 simplr 769 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑦𝑆)
48 simpr 489 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑧𝑆)
4947, 48ovresd 7312 . . . . . . . . . 10 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑦𝐻𝑧))
501, 2, 8, 25, 28homfval 17013 . . . . . . . . . 10 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑦𝐻𝑧) = (𝑦(Hom ‘𝐶)𝑧))
5149, 50eqtrd 2794 . . . . . . . . 9 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑦(Hom ‘𝐶)𝑧))
5251raleqdv 3330 . . . . . . . 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 3320 . . . . . . 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 260 . . . . . 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 3114 . . . . 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 3114 . . . 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 516 . . 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 3114 . 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 5540 . . . . 5 ((𝑆𝐵𝑆𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
6012, 12, 59syl2anc 588 . . . 4 (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
61 fnssres 6454 . . . 4 ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
623, 60, 61sylancr 591 . . 3 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
631, 9, 20, 10, 62issubc2 17158 . 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 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071  Vcvv 3410   ⊆ wss 3859  ⟨cop 4529   class class class wbr 5033   × cxp 5523   ↾ cres 5527   Fn wfn 6331  ‘cfv 6336  (class class class)co 7151  Basecbs 16534  Hom chom 16627  compcco 16628  Catccat 16986  Idccid 16987  Homf chomf 16988   ⊆cat cssc 17129  Subcatcsubc 17131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7694  df-2nd 7695  df-pm 8420  df-ixp 8481  df-cat 16990  df-cid 16991  df-homf 16992  df-ssc 17132  df-subc 17134 This theorem is referenced by:  resscat  17174  funcres2c  17223  ressffth  17260  funcsetcres2  17412
 Copyright terms: Public domain W3C validator