Theorem subccocl 17803

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 2737	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
3		eqid 2737	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		subccocl.o	. . . . 5 ⊢ · = (comp‘𝐶)
5		subcrcl 17774	. . . . . 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 17794	. . . 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 232	. . 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 495	. 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑌 ∈ 𝑆)
14		subccocl.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑆)
15	14	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) → 𝑍 ∈ 𝑆)
16		subccocl.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌))
17	16	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑋𝐽𝑌))
18		simpllr 776	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑥 = 𝑋)
19		simplr 769	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝑦 = 𝑌)
20	18, 19	oveq12d 7378	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → (𝑥𝐽𝑦) = (𝑋𝐽𝑌))
21	17, 20	eleqtrrd 2840	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) → 𝐹 ∈ (𝑥𝐽𝑦))
22		subccocl.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐽𝑍))
23	22	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝐺 ∈ (𝑌𝐽𝑍))
24		simpllr 776	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝑦 = 𝑌)
25		simplr 769	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝑧 = 𝑍)
26	24, 25	oveq12d 7378	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → (𝑦𝐽𝑧) = (𝑌𝐽𝑍))
27	23, 26	eleqtrrd 2840	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) → 𝐺 ∈ (𝑦𝐽𝑧))
28		simp-5r 786	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑥 = 𝑋)
29		simp-4r 784	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑦 = 𝑌)
30	28, 29	opeq12d 4825	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
31		simpllr 776	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑧 = 𝑍)
32	30, 31	oveq12d 7378	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (⟨𝑥, 𝑦⟩ · 𝑧) = (⟨𝑋, 𝑌⟩ · 𝑍))
33		simpr 484	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
34		simplr 769	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
35	32, 33, 34	oveq123d 7381	. . . . . . . . 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 7378	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑌) ∧ 𝑧 = 𝑍) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑥𝐽𝑧) = (𝑋𝐽𝑍))
37	35, 36	eleq12d 2831	. . . . . . . 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 3557	. . . . . . 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 3555	. . . . . 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 3555	. . . . 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 3555	. . . 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 490	. . 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 3555	. 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 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ⟨cop 4574   class class class wbr 5086   × cxp 5622   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360  compcco 17223  Catccat 17621  Idccid 17622  Hom_f chomf 17623   ⊆_cat cssc 17765  Subcatcsubc 17767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-pm 8769  df-ixp 8839  df-ssc 17768  df-subc 17770

This theorem is referenced by:  subccatid  17804  funcres  17854  iinfsubc  49545