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Theorem fullsubc 16711
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 16554 . . . 4 𝐻 Fn (𝐵 × 𝐵)
4 fvex 6362 . . . . 5 (Base‘𝐶) ∈ V
52, 4eqeltri 2835 . . . 4 𝐵 ∈ V
6 sscres 16684 . . . 4 ((𝐻 Fn (𝐵 × 𝐵) ∧ 𝐵 ∈ V) → (𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻)
73, 5, 6mp2an 710 . . 3 (𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻
87a1i 11 . 2 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) ⊆cat 𝐻)
9 eqid 2760 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
10 eqid 2760 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
11 fullsubc.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
1211adantr 472 . . . . . 6 ((𝜑𝑥𝑆) → 𝐶 ∈ Cat)
13 fullsubc.s . . . . . . 7 (𝜑𝑆𝐵)
1413sselda 3744 . . . . . 6 ((𝜑𝑥𝑆) → 𝑥𝐵)
152, 9, 10, 12, 14catidcl 16544 . . . . 5 ((𝜑𝑥𝑆) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
16 simpr 479 . . . . . . 7 ((𝜑𝑥𝑆) → 𝑥𝑆)
1716, 16ovresd 6966 . . . . . 6 ((𝜑𝑥𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) = (𝑥𝐻𝑥))
181, 2, 9, 14, 14homfval 16553 . . . . . 6 ((𝜑𝑥𝑆) → (𝑥𝐻𝑥) = (𝑥(Hom ‘𝐶)𝑥))
1917, 18eqtrd 2794 . . . . 5 ((𝜑𝑥𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥) = (𝑥(Hom ‘𝐶)𝑥))
2015, 19eleqtrrd 2842 . . . 4 ((𝜑𝑥𝑆) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑥))
21 eqid 2760 . . . . . . . . . 10 (comp‘𝐶) = (comp‘𝐶)
2212ad3antrrr 768 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐶 ∈ Cat)
2314ad3antrrr 768 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥𝐵)
2413adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑆𝐵)
2524sselda 3744 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑦𝐵)
2625adantr 472 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑦𝐵)
2726adantr 472 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦𝐵)
2824adantr 472 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑆𝐵)
2928sselda 3744 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑧𝐵)
3029adantr 472 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝐵)
31 simprl 811 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
32 simprr 813 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
332, 9, 21, 22, 23, 27, 30, 31, 32catcocl 16547 . . . . . . . . 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))
3416ad3antrrr 768 . . . . . . . . . . 11 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥𝑆)
35 simplr 809 . . . . . . . . . . 11 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧𝑆)
3634, 35ovresd 6966 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑥𝐻𝑧))
371, 2, 9, 23, 30homfval 16553 . . . . . . . . . 10 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥𝐻𝑧) = (𝑥(Hom ‘𝐶)𝑧))
3836, 37eqtrd 2794 . . . . . . . . 9 (((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑥(Hom ‘𝐶)𝑧))
3933, 38eleqtrrd 2842 . . . . . . . 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))
4039ralrimivva 3109 . . . . . . 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))
41 simplr 809 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑥𝑆)
42 simpr 479 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑦𝑆)
4341, 42ovresd 6966 . . . . . . . . . 10 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥𝐻𝑦))
4414adantr 472 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → 𝑥𝐵)
451, 2, 9, 44, 25homfval 16553 . . . . . . . . . 10 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
4643, 45eqtrd 2794 . . . . . . . . 9 (((𝜑𝑥𝑆) ∧ 𝑦𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐶)𝑦))
4746adantr 472 . . . . . . . 8 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐶)𝑦))
48 simplr 809 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑦𝑆)
49 simpr 479 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → 𝑧𝑆)
5048, 49ovresd 6966 . . . . . . . . . 10 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑦𝐻𝑧))
511, 2, 9, 26, 29homfval 16553 . . . . . . . . . 10 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑦𝐻𝑧) = (𝑦(Hom ‘𝐶)𝑧))
5250, 51eqtrd 2794 . . . . . . . . 9 ((((𝜑𝑥𝑆) ∧ 𝑦𝑆) ∧ 𝑧𝑆) → (𝑦(𝐻 ↾ (𝑆 × 𝑆))𝑧) = (𝑦(Hom ‘𝐶)𝑧))
5352raleqdv 3283 . . . . . . . 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)))
5447, 53raleqbidv 3291 . . . . . . 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)))
5540, 54mpbird 247 . . . . . 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))
5655ralrimiva 3104 . . . . 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))
5756ralrimiva 3104 . . . 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))
5820, 57jca 555 . . 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)))
5958ralrimiva 3104 . 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)))
60 xpss12 5281 . . . . 5 ((𝑆𝐵𝑆𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
6113, 13, 60syl2anc 696 . . . 4 (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
62 fnssres 6165 . . . 4 ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
633, 61, 62sylancr 698 . . 3 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
641, 10, 21, 11, 63issubc2 16697 . 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)))))
658, 59, 64mpbir2and 995 1 (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) ∈ (Subcat‘𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  wss 3715  cop 4327   class class class wbr 4804   × cxp 5264  cres 5268   Fn wfn 6044  cfv 6049  (class class class)co 6813  Basecbs 16059  Hom chom 16154  compcco 16155  Catccat 16526  Idccid 16527  Homf chomf 16528  cat cssc 16668  Subcatcsubc 16670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-pm 8026  df-ixp 8075  df-cat 16530  df-cid 16531  df-homf 16532  df-ssc 16671  df-subc 16673
This theorem is referenced by:  resscat  16713  funcres2c  16762  ressffth  16799  funcsetcres2  16944
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