Theorem subccocl 17109

Description: A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
subcidcl.j	⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
subcidcl.2	⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
subcidcl.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)
subccocl.o	⊢ · = (comp‘𝐶)
subccocl.y	⊢ (𝜑 → 𝑌 ∈ 𝑆)
subccocl.z	⊢ (𝜑 → 𝑍 ∈ 𝑆)
subccocl.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌))
subccocl.g	⊢ (𝜑 → 𝐺 ∈ (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.

Step	Hyp	Ref	Expression
1		subcidcl.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶))
2		eqid 2821	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
3		eqid 2821	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		subccocl.o	. . . . 5 ⊢ · = (comp‘𝐶)
5		subcrcl 17080	. . . . . 6 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
6	1, 5	syl 17	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		subcidcl.2	. . . . 5 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆))
8	2, 3, 4, 6, 7	issubc2 17100	. . . 4 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ 𝑆 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))))
9	1, 8	mpbid 234	. . 3 ⊢ (𝜑 → (𝐽 ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ 𝑆 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧))))
10	9	simprd 498	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)))
11		subcidcl.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
12		subccocl.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
13	12	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝑆)
14		subccocl.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑆)
15	14	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑍 ∈ 𝑆)
16		subccocl.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌))
17	16	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑋𝐽𝑌))
18		simpllr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑥 = 𝑋)
19		simplr 767	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑦 = 𝑌)
20	18, 19	oveq12d 7168	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑥𝐽𝑦) = (𝑋𝐽𝑌))
21	17, 20	eleqtrrd 2916	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑥𝐽𝑦))
22		subccocl.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐽𝑍))
23	22	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝐺 ∈ (𝑌𝐽𝑍))
24		simpllr 774	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝑦 = 𝑌)
25		simplr 767	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝑧 = 𝑍)
26	24, 25	oveq12d 7168	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → (𝑦𝐽𝑧) = (𝑌𝐽𝑍))
27	23, 26	eleqtrrd 2916	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝐺 ∈ (𝑦𝐽𝑧))
28		simp-5r 784	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑥 = 𝑋)
29		simp-4r 782	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑦 = 𝑌)
30	28, 29	opeq12d 4804	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
31		simpllr 774	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑧 = 𝑍)
32	30, 31	oveq12d 7168	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (⟨𝑥, 𝑦⟩ · 𝑧) = (⟨𝑋, 𝑌⟩ · 𝑍))
33		simpr 487	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
34		simplr 767	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
35	32, 33, 34	oveq123d 7171	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
36	28, 31	oveq12d 7168	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑥𝐽𝑧) = (𝑋𝐽𝑍))
37	35, 36	eleq12d 2907	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧) ↔ (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
38	27, 37	rspcdv 3614	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → (∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
39	21, 38	rspcimdv 3612	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
40	15, 39	rspcimdv 3612	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → (∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
41	13, 40	rspcimdv 3612	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
42	41	adantld 493	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
43	11, 42	rspcimdv 3612	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐽𝑍)))
44	10, 43	mpd 15	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐽𝑍))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ⟨cop 4566   class class class wbr 5058   × cxp 5547   Fn wfn 6344  ‘cfv 6349  (class class class)co 7150  compcco 16571  Catccat 16929  Idccid 16930  Hom_f chomf 16931   ⊆_cat cssc 17071  Subcatcsubc 17073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-pm 8403  df-ixp 8456  df-ssc 17074  df-subc 17076

This theorem is referenced by:  subccatid  17110  funcres  17160