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

Theorem subccocl 16726
 Description: A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subcidcl.j (𝜑𝐽 ∈ (Subcat‘𝐶))
subcidcl.2 (𝜑𝐽 Fn (𝑆 × 𝑆))
subcidcl.x (𝜑𝑋𝑆)
subccocl.o · = (comp‘𝐶)
subccocl.y (𝜑𝑌𝑆)
subccocl.z (𝜑𝑍𝑆)
subccocl.f (𝜑𝐹 ∈ (𝑋𝐽𝑌))
subccocl.g (𝜑𝐺 ∈ (𝑌𝐽𝑍))
Assertion
Ref Expression
subccocl (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐽𝑍))

Proof of Theorem subccocl
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subcidcl.j . . . 4 (𝜑𝐽 ∈ (Subcat‘𝐶))
2 eqid 2760 . . . . 5 (Homf𝐶) = (Homf𝐶)
3 eqid 2760 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
4 subccocl.o . . . . 5 · = (comp‘𝐶)
5 subcrcl 16697 . . . . . 6 (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
61, 5syl 17 . . . . 5 (𝜑𝐶 ∈ Cat)
7 subcidcl.2 . . . . 5 (𝜑𝐽 Fn (𝑆 × 𝑆))
82, 3, 4, 6, 7issubc2 16717 . . . 4 (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat (Homf𝐶) ∧ ∀𝑥𝑆 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
91, 8mpbid 222 . . 3 (𝜑 → (𝐽cat (Homf𝐶) ∧ ∀𝑥𝑆 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
109simprd 482 . 2 (𝜑 → ∀𝑥𝑆 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
11 subcidcl.x . . 3 (𝜑𝑋𝑆)
12 subccocl.y . . . . . 6 (𝜑𝑌𝑆)
1312adantr 472 . . . . 5 ((𝜑𝑥 = 𝑋) → 𝑌𝑆)
14 subccocl.z . . . . . . 7 (𝜑𝑍𝑆)
1514ad2antrr 764 . . . . . 6 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑍𝑆)
16 subccocl.f . . . . . . . . 9 (𝜑𝐹 ∈ (𝑋𝐽𝑌))
1716ad3antrrr 768 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑋𝐽𝑌))
18 simpllr 817 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑥 = 𝑋)
19 simplr 809 . . . . . . . . 9 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑦 = 𝑌)
2018, 19oveq12d 6832 . . . . . . . 8 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑥𝐽𝑦) = (𝑋𝐽𝑌))
2117, 20eleqtrrd 2842 . . . . . . 7 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑥𝐽𝑦))
22 subccocl.g . . . . . . . . . 10 (𝜑𝐺 ∈ (𝑌𝐽𝑍))
2322ad4antr 771 . . . . . . . . 9 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝐺 ∈ (𝑌𝐽𝑍))
24 simpllr 817 . . . . . . . . . 10 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝑦 = 𝑌)
25 simplr 809 . . . . . . . . . 10 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝑧 = 𝑍)
2624, 25oveq12d 6832 . . . . . . . . 9 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → (𝑦𝐽𝑧) = (𝑌𝐽𝑍))
2723, 26eleqtrrd 2842 . . . . . . . 8 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝐺 ∈ (𝑦𝐽𝑧))
28 simp-5r 831 . . . . . . . . . . . 12 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑥 = 𝑋)
29 simp-4r 827 . . . . . . . . . . . 12 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑦 = 𝑌)
3028, 29opeq12d 4561 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
31 simpllr 817 . . . . . . . . . . 11 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑧 = 𝑍)
3230, 31oveq12d 6832 . . . . . . . . . 10 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (⟨𝑥, 𝑦· 𝑧) = (⟨𝑋, 𝑌· 𝑍))
33 simpr 479 . . . . . . . . . 10 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
34 simplr 809 . . . . . . . . . 10 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
3532, 33, 34oveq123d 6835 . . . . . . . . 9 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
3628, 31oveq12d 6832 . . . . . . . . 9 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑥𝐽𝑧) = (𝑋𝐽𝑍))
3735, 36eleq12d 2833 . . . . . . . 8 ((((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧) ↔ (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
3827, 37rspcdv 3452 . . . . . . 7 (((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → (∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
3921, 38rspcimdv 3450 . . . . . 6 ((((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
4015, 39rspcimdv 3450 . . . . 5 (((𝜑𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (∀𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
4113, 40rspcimdv 3450 . . . 4 ((𝜑𝑥 = 𝑋) → (∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
4241adantld 484 . . 3 ((𝜑𝑥 = 𝑋) → ((((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
4311, 42rspcimdv 3450 . 2 (𝜑 → (∀𝑥𝑆 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦𝑆𝑧𝑆𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦· 𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
4410, 43mpd 15 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) ∈ (𝑋𝐽𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ⟨cop 4327   class class class wbr 4804   × cxp 5264   Fn wfn 6044  ‘cfv 6049  (class class class)co 6814  compcco 16175  Catccat 16546  Idccid 16547  Homf chomf 16548   ⊆cat cssc 16688  Subcatcsubc 16690 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 7115 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-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-ov 6817  df-oprab 6818  df-mpt2 6819  df-pm 8028  df-ixp 8077  df-ssc 16691  df-subc 16693 This theorem is referenced by:  subccatid  16727  funcres  16777
 Copyright terms: Public domain W3C validator