Theorem istopclsd 43553

Description: A closure function which satisfies sscls 23287, clsidm 23298, cls0 23311, and clsun 36955 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses

Ref	Expression
istopclsd.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
istopclsd.f	⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
istopclsd.e	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
istopclsd.i	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))
istopclsd.z	⊢ (𝜑 → (𝐹‘∅) = ∅)
istopclsd.u	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝐹‘(𝑥 ∪ 𝑦)) = ((𝐹‘𝑥) ∪ (𝐹‘𝑦)))
istopclsd.j	⊢ 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)}

Assertion

Ref	Expression
istopclsd	⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹))

Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐽,𝑦   𝑥,𝑉,𝑦,𝑧

Allowed substitution hint:   𝐽(𝑧)

Proof of Theorem istopclsd

Step	Hyp	Ref	Expression
1		istopclsd.j	. . . 4 ⊢ 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)}
2		istopclsd.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
3	2	ffnd 6707	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝒫 𝐵)
4	3	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → 𝐹 Fn 𝒫 𝐵)
5		difss 4086	. . . . . . . . 9 ⊢ (𝐵 ∖ 𝑧) ⊆ 𝐵
6		istopclsd.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		elpw2g 5302	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑉 → ((𝐵 ∖ 𝑧) ∈ 𝒫 𝐵 ↔ (𝐵 ∖ 𝑧) ⊆ 𝐵))
8	6, 7	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 ∖ 𝑧) ∈ 𝒫 𝐵 ↔ (𝐵 ∖ 𝑧) ⊆ 𝐵))
9	5, 8	mpbiri 261	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∖ 𝑧) ∈ 𝒫 𝐵)
10	9	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑧) ∈ 𝒫 𝐵)
11		fnelfp 7177	. . . . . . 7 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ (𝐵 ∖ 𝑧) ∈ 𝒫 𝐵) → ((𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)))
12	4, 10, 11	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → ((𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)))
13	12	bicomd 226	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → ((𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧) ↔ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )))
14	13	rabbidva 3420	. . . 4 ⊢ (𝜑 → {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )})
15	1, 14	eqtrid 2809	. . 3 ⊢ (𝜑 → 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )})
16		istopclsd.e	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
17		simp1 1154	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → 𝜑)
18		simp2 1155	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → 𝑥 ⊆ 𝐵)
19		simp3 1156	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → 𝑦 ⊆ 𝑥)
20	19, 18	sstrd 3944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → 𝑦 ⊆ 𝐵)
21		istopclsd.u	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝐹‘(𝑥 ∪ 𝑦)) = ((𝐹‘𝑥) ∪ (𝐹‘𝑦)))
22	17, 18, 20, 21	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘(𝑥 ∪ 𝑦)) = ((𝐹‘𝑥) ∪ (𝐹‘𝑦)))
23		ssequn2 4138	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑥 ↔ (𝑥 ∪ 𝑦) = 𝑥)
24	23	biimpi 219	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ∪ 𝑦) = 𝑥)
25	24	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∪ 𝑦) = 𝑥)
26	25	fveq2d 6886	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘(𝑥 ∪ 𝑦)) = (𝐹‘𝑥))
27	22, 26	eqtr3d 2799	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → ((𝐹‘𝑥) ∪ (𝐹‘𝑦)) = (𝐹‘𝑥))
28		ssequn2 4138	. . . . . . 7 ⊢ ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ ((𝐹‘𝑥) ∪ (𝐹‘𝑦)) = (𝐹‘𝑥))
29	27, 28	sylibr 237	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥))
30		istopclsd.i	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))
31	6, 2, 16, 29, 30	ismrcd1 43551	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
32		istopclsd.z	. . . . . 6 ⊢ (𝜑 → (𝐹‘∅) = ∅)
33		0elpw 5324	. . . . . . 7 ⊢ ∅ ∈ 𝒫 𝐵
34		fnelfp 7177	. . . . . . 7 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) → (∅ ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∅) = ∅))
35	3, 33, 34	sylancl 598	. . . . . 6 ⊢ (𝜑 → (∅ ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∅) = ∅))
36	32, 35	mpbird 260	. . . . 5 ⊢ (𝜑 → ∅ ∈ dom (𝐹 ∩ I ))
37		simp1 1154	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝜑)
38		inss1 4185	. . . . . . . . . . . . 13 ⊢ (𝐹 ∩ I ) ⊆ 𝐹
39		dmss 5890	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∩ I ) ⊆ 𝐹 → dom (𝐹 ∩ I ) ⊆ dom 𝐹)
40	38, 39	ax-mp 5	. . . . . . . . . . . 12 ⊢ dom (𝐹 ∩ I ) ⊆ dom 𝐹
41	40, 2	fssdm 6726	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
42	41	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
43		simp2 1155	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ∈ dom (𝐹 ∩ I ))
44	42, 43	sseldd 3935	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ∈ 𝒫 𝐵)
45	44	elpwid 4569	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ⊆ 𝐵)
46		simp3 1156	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ∈ dom (𝐹 ∩ I ))
47	42, 46	sseldd 3935	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ∈ 𝒫 𝐵)
48	47	elpwid 4569	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ⊆ 𝐵)
49	37, 45, 48, 21	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘(𝑥 ∪ 𝑦)) = ((𝐹‘𝑥) ∪ (𝐹‘𝑦)))
50	3	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝐹 Fn 𝒫 𝐵)
51		fnelfp 7177	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝑥 ∈ 𝒫 𝐵) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
52	50, 44, 51	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
53	43, 52	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘𝑥) = 𝑥)
54		fnelfp 7177	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝑦 ∈ 𝒫 𝐵) → (𝑦 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑦) = 𝑦))
55	50, 47, 54	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑦 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑦) = 𝑦))
56	46, 55	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘𝑦) = 𝑦)
57	53, 56	uneq12d 4119	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → ((𝐹‘𝑥) ∪ (𝐹‘𝑦)) = (𝑥 ∪ 𝑦))
58	49, 57	eqtrd 2797	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘(𝑥 ∪ 𝑦)) = (𝑥 ∪ 𝑦))
59	45, 48	unssd 4141	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥 ∪ 𝑦) ⊆ 𝐵)
60		vex 3457	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
61		vex 3457	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
62	60, 61	unex 7750	. . . . . . . . 9 ⊢ (𝑥 ∪ 𝑦) ∈ V
63	62	elpw 4564	. . . . . . . 8 ⊢ ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐵 ↔ (𝑥 ∪ 𝑦) ⊆ 𝐵)
64	59, 63	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥 ∪ 𝑦) ∈ 𝒫 𝐵)
65		fnelfp 7177	. . . . . . 7 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ (𝑥 ∪ 𝑦) ∈ 𝒫 𝐵) → ((𝑥 ∪ 𝑦) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝑥 ∪ 𝑦)) = (𝑥 ∪ 𝑦)))
66	50, 64, 65	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → ((𝑥 ∪ 𝑦) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝑥 ∪ 𝑦)) = (𝑥 ∪ 𝑦)))
67	58, 66	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥 ∪ 𝑦) ∈ dom (𝐹 ∩ I ))
68		eqid 2762	. . . . 5 ⊢ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )}
69	31, 36, 67, 68	mretopd 23323	. . . 4 ⊢ (𝜑 → ({𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )} ∈ (TopOn‘𝐵) ∧ dom (𝐹 ∩ I ) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )})))
70	69	simpld 500	. . 3 ⊢ (𝜑 → {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )} ∈ (TopOn‘𝐵))
71	15, 70	eqeltrd 2862	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵))
72		topontop 23144	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
73	71, 72	syl 18	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
74		eqid 2762	. . . . . 6 ⊢ (mrCls‘(Clsd‘𝐽)) = (mrCls‘(Clsd‘𝐽))
75	74	mrccls 23310	. . . . 5 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (mrCls‘(Clsd‘𝐽)))
76	73, 75	syl 18	. . . 4 ⊢ (𝜑 → (cls‘𝐽) = (mrCls‘(Clsd‘𝐽)))
77	69	simprd 501	. . . . . 6 ⊢ (𝜑 → dom (𝐹 ∩ I ) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )}))
78	15	fveq2d 6886	. . . . . 6 ⊢ (𝜑 → (Clsd‘𝐽) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )}))
79	77, 78	eqtr4d 2800	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ I ) = (Clsd‘𝐽))
80	79	fveq2d 6886	. . . 4 ⊢ (𝜑 → (mrCls‘dom (𝐹 ∩ I )) = (mrCls‘(Clsd‘𝐽)))
81	76, 80	eqtr4d 2800	. . 3 ⊢ (𝜑 → (cls‘𝐽) = (mrCls‘dom (𝐹 ∩ I )))
82	6, 2, 16, 29, 30	ismrcd2 43552	. . 3 ⊢ (𝜑 → 𝐹 = (mrCls‘dom (𝐹 ∩ I )))
83	81, 82	eqtr4d 2800	. 2 ⊢ (𝜑 → (cls‘𝐽) = 𝐹)
84	71, 83	jca 521	1 ⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3414   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560   I cid 5553  dom cdm 5659   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  mrClscmrc 17673  Topctop 23124  TopOnctopon 23141  Clsdccld 23247  clsccl 23249

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-mre 17676  df-mrc 17677  df-top 23125  df-topon 23142  df-cld 23250  df-cls 23252

This theorem is used by: (None)