Theorem parcatid 50969

Description: Construct a category with a family of parallel arrows from 𝑋 to 𝑌 and only identity endomorphisms. All arrow values are subsets of 𝐽; composition is intersection. Nonidentity arrows can occur only when 𝑋 and 𝑌 are distinct members of 𝐵. If 𝑋 = 𝑌, or either distinguished endpoint is outside 𝐵, the category is discrete. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)

Hypotheses

Ref	Expression
parcatid.b	⊢ 𝐵 ∈ V
parcatid.j	⊢ 𝐽 ∈ V
parcatid.p	⊢ 𝑆 ⊆ 𝒫 𝐽
parcatid.k	⊢ 𝐾 = (𝑆 ∪ {𝐽})
parcatid.h	⊢ 𝐻 = (𝑚 ∈ 𝐵, 𝑛 ∈ 𝐵 ↦ if(𝑚 = 𝑛, {𝐽}, if((𝑚 = 𝑋 ∧ 𝑛 = 𝑌), 𝑆, ∅)))
parcatid.m	⊢ 𝑀 = (𝑟 ∈ 𝐾, 𝑠 ∈ 𝐾 ↦ (𝑟 ∩ 𝑠))
parcatid.o	⊢ · = (𝑝 ∈ (𝐵 × 𝐵), 𝑞 ∈ 𝐵 ↦ 𝑀)
parcatid.c	⊢ 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩}

Assertion

Ref	Expression
parcatid	⊢ (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐽))

Distinct variable groups:   𝐵,𝑚,𝑛,𝑟,𝑠,𝑝,𝑞,𝑦   𝑚,𝐽,𝑛,𝑟,𝑠,𝑝,𝑞,𝑦   𝑆,𝑚,𝑛,𝑟,𝑠,𝑝,𝑞,𝑦   𝑚,𝑋,𝑛,𝑟,𝑠,𝑝,𝑞,𝑦   𝑚,𝑌,𝑛,𝑟,𝑠,𝑝,𝑞,𝑦   𝑚,𝐾,𝑛,𝑟,𝑠,𝑝,𝑞,𝑦   𝐻,𝑝,𝑞,𝑟,𝑠,𝑦   𝑚,𝑀,𝑛,𝑝,𝑞,𝑦   𝑦,𝐶   𝑦, ·

Allowed substitution hints:   𝐶(𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝)   · (𝑚, 𝑛, 𝑠, 𝑟, 𝑞, 𝑝)   𝐻(𝑚, 𝑛)   𝑀(𝑠, 𝑟)

Proof of Theorem parcatid

Dummy variables 𝑥 𝑧 𝑤 𝑓 𝑔 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		parcatid.c	. . . . 5 ⊢ 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩}
2		parcatid.b	. . . . 5 ⊢ 𝐵 ∈ V
3	1, 2	catbas 50553	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4	3	a1i 11	. . 3 ⊢ (⊤ → 𝐵 = (Base‘𝐶))
5		parcatid.h	. . . . . 6 ⊢ 𝐻 = (𝑚 ∈ 𝐵, 𝑛 ∈ 𝐵 ↦ if(𝑚 = 𝑛, {𝐽}, if((𝑚 = 𝑋 ∧ 𝑛 = 𝑌), 𝑆, ∅)))
6	2, 2	mpoex 8117	. . . . . 6 ⊢ (𝑚 ∈ 𝐵, 𝑛 ∈ 𝐵 ↦ if(𝑚 = 𝑛, {𝐽}, if((𝑚 = 𝑋 ∧ 𝑛 = 𝑌), 𝑆, ∅))) ∈ V
7	5, 6	eqeltri 2860	. . . . 5 ⊢ 𝐻 ∈ V
8	1, 7	cathomfval 50554	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
9	8	a1i 11	. . 3 ⊢ (⊤ → 𝐻 = (Hom ‘𝐶))
10		parcatid.o	. . . . . 6 ⊢ · = (𝑝 ∈ (𝐵 × 𝐵), 𝑞 ∈ 𝐵 ↦ 𝑀)
11	2, 2	xpex 7791	. . . . . . 7 ⊢ (𝐵 × 𝐵) ∈ V
12	11, 2	mpoex 8117	. . . . . 6 ⊢ (𝑝 ∈ (𝐵 × 𝐵), 𝑞 ∈ 𝐵 ↦ 𝑀) ∈ V
13	10, 12	eqeltri 2860	. . . . 5 ⊢ · ∈ V
14	1, 13	catcofval 50555	. . . 4 ⊢ · = (comp‘𝐶)
15	14	a1i 11	. . 3 ⊢ (⊤ → · = (comp‘𝐶))
16		tpex 7786	. . . . 5 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V
17	1, 16	eqeltri 2860	. . . 4 ⊢ 𝐶 ∈ V
18	17	a1i 11	. . 3 ⊢ (⊤ → 𝐶 ∈ V)
19		biid 264	. . 3 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
20		parcatid.j	. . . 4 ⊢ 𝐽 ∈ V
21		simpr 490	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
22	5, 20, 21	parhomid 50957	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐵) → 𝐽 ∈ (𝑦𝐻𝑦))
23		parcatid.m	. . . . 5 ⊢ 𝑀 = (𝑟 ∈ 𝐾, 𝑠 ∈ 𝐾 ↦ (𝑟 ∩ 𝑠))
24		parcatid.k	. . . . . . . 8 ⊢ 𝐾 = (𝑆 ∪ {𝐽})
25	20	pwex 5346	. . . . . . . . . 10 ⊢ 𝒫 𝐽 ∈ V
26		parcatid.p	. . . . . . . . . 10 ⊢ 𝑆 ⊆ 𝒫 𝐽
27	25, 26	ssexi 5288	. . . . . . . . 9 ⊢ 𝑆 ∈ V
28		snex 5401	. . . . . . . . 9 ⊢ {𝐽} ∈ V
29	27, 28	unex 7785	. . . . . . . 8 ⊢ (𝑆 ∪ {𝐽}) ∈ V
30	24, 29	eqeltri 2860	. . . . . . 7 ⊢ 𝐾 ∈ V
31	30, 30	mpoex 8117	. . . . . 6 ⊢ (𝑟 ∈ 𝐾, 𝑠 ∈ 𝐾 ↦ (𝑟 ∩ 𝑠)) ∈ V
32	23, 31	eqeltri 2860	. . . . 5 ⊢ 𝑀 ∈ V
33		simpr 490	. . . . . . 7 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤))))
34	33	simp1d 1160	. . . . . 6 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
35	34	simpld 500	. . . . 5 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑥 ∈ 𝐵)
36	34	simprd 501	. . . . 5 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑦 ∈ 𝐵)
37	33	simp3d 1162	. . . . . . 7 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))
38	37	simp1d 1160	. . . . . 6 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑓 ∈ (𝑥𝐻𝑦))
39	5, 24, 38	parhomcl 50953	. . . . 5 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑓 ∈ 𝐾)
40	20	snid 4623	. . . . . . . 8 ⊢ 𝐽 ∈ {𝐽}
41		elun2 4184	. . . . . . . 8 ⊢ (𝐽 ∈ {𝐽} → 𝐽 ∈ (𝑆 ∪ {𝐽}))
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ 𝐽 ∈ (𝑆 ∪ {𝐽})
43	42, 24	eleqtrri 2863	. . . . . 6 ⊢ 𝐽 ∈ 𝐾
44	43	a1i 11	. . . . 5 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝐽 ∈ 𝐾)
45	23, 10, 32, 35, 36, 36, 39, 44	parcoval 50967	. . . 4 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝐽(⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = (𝐽 ∩ 𝑓))
46	5, 26, 38	parhomss 50962	. . . . 5 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑓 ⊆ 𝐽)
47		sseqin2 4123	. . . . 5 ⊢ (𝑓 ⊆ 𝐽 ↔ (𝐽 ∩ 𝑓) = 𝑓)
48	46, 47	sylib 221	. . . 4 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝐽 ∩ 𝑓) = 𝑓)
49	45, 48	eqtrd 2798	. . 3 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝐽(⟨𝑥, 𝑦⟩ · 𝑦)𝑓) = 𝑓)
50	33	simp2d 1161	. . . . . 6 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))
51	50	simpld 500	. . . . 5 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑧 ∈ 𝐵)
52	37	simp2d 1161	. . . . . 6 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑔 ∈ (𝑦𝐻𝑧))
53	5, 24, 52	parhomcl 50953	. . . . 5 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑔 ∈ 𝐾)
54	23, 10, 32, 36, 36, 51, 44, 53	parcoval 50967	. . . 4 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧)𝐽) = (𝑔 ∩ 𝐽))
55	5, 26, 52	parhomss 50962	. . . . 5 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑔 ⊆ 𝐽)
56		dfss2 4097	. . . . 5 ⊢ (𝑔 ⊆ 𝐽 ↔ (𝑔 ∩ 𝐽) = 𝑔)
57	55, 56	sylib 221	. . . 4 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑔 ∩ 𝐽) = 𝑔)
58	54, 57	eqtrd 2798	. . 3 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑔(⟨𝑦, 𝑦⟩ · 𝑧)𝐽) = 𝑔)
59	23, 10, 32, 35, 36, 51, 39, 53	parcoval 50967	. . . 4 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) = (𝑔 ∩ 𝑓))
60	5, 38, 52, 26	parhomco 50966	. . . 4 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑔 ∩ 𝑓) ∈ (𝑥𝐻𝑧))
61	59, 60	eqeltrd 2864	. . 3 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))
62		inass 4127	. . . 4 ⊢ ((𝑘 ∩ 𝑔) ∩ 𝑓) = (𝑘 ∩ (𝑔 ∩ 𝑓))
63	50	simprd 501	. . . . . . 7 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑤 ∈ 𝐵)
64	37	simp3d 1162	. . . . . . . 8 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑘 ∈ (𝑧𝐻𝑤))
65	5, 24, 64	parhomcl 50953	. . . . . . 7 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → 𝑘 ∈ 𝐾)
66	23, 10, 32, 36, 51, 63, 53, 65	parcoval 50967	. . . . . 6 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔) = (𝑘 ∩ 𝑔))
67	66	oveq1d 7458	. . . . 5 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = ((𝑘 ∩ 𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓))
68	5, 52, 64, 26	parhomco 50966	. . . . . . 7 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑘 ∩ 𝑔) ∈ (𝑦𝐻𝑤))
69	5, 24, 68	parhomcl 50953	. . . . . 6 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑘 ∩ 𝑔) ∈ 𝐾)
70	23, 10, 32, 35, 36, 63, 39, 69	parcoval 50967	. . . . 5 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → ((𝑘 ∩ 𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = ((𝑘 ∩ 𝑔) ∩ 𝑓))
71	67, 70	eqtrd 2798	. . . 4 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = ((𝑘 ∩ 𝑔) ∩ 𝑓))
72	59	oveq2d 7459	. . . . 5 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔 ∩ 𝑓)))
73	5, 24, 60	parhomcl 50953	. . . . . 6 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑔 ∩ 𝑓) ∈ 𝐾)
74	23, 10, 32, 35, 51, 63, 73, 65	parcoval 50967	. . . . 5 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔 ∩ 𝑓)) = (𝑘 ∩ (𝑔 ∩ 𝑓)))
75	72, 74	eqtrd 2798	. . . 4 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)) = (𝑘 ∩ (𝑔 ∩ 𝑓)))
76	62, 71, 75	3eqtr4a 2824	. . 3 ⊢ ((⊤ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧) ∧ 𝑘 ∈ (𝑧𝐻𝑤)))) → ((𝑘(⟨𝑦, 𝑧⟩ · 𝑤)𝑔)(⟨𝑥, 𝑦⟩ · 𝑤)𝑓) = (𝑘(⟨𝑥, 𝑧⟩ · 𝑤)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓)))
77	4, 9, 15, 18, 19, 22, 49, 58, 61, 76	iscatd2 17884	. 2 ⊢ (⊤ → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐽)))
78	77	mptru 1577	1 ⊢ (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐽))

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  ifcif 4485  𝒫 cpw 4560  {csn 4588  {ctp 4592  ⟨cop 4838   ↦ cmpt 5189   × cxp 5653  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  ndxcnx 17400  Basecbs 17416  Hom chom 17468  compcco 17469  Catccat 17867  Idccid 17868

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-struct 17354  df-slot 17389  df-ndx 17401  df-base 17417  df-hom 17481  df-cco 17482  df-cat 17871  df-cid 17872

This theorem is used by:  par2catid  50972