Theorem istopclsd 43804

Description: A closure function which satisfies sscls 23379, clsidm 23390, cls0 23403, and clsun 37208 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 6710	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝒫 𝐵)
4	3	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → 𝐹 Fn 𝒫 𝐵)
5		difss 4072	. . . . . . . . 9 ⊢ (𝐵 ∖ 𝑧) ⊆ 𝐵
6		istopclsd.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		elpw2g 5294	. . . . . . . . . 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 7180	. . . . . . 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 3418	. . . 4 ⊢ (𝜑 → {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )})
15	1, 14	eqtrid 2807	. . 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 3918	. . . . . . . . 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 4187	. . . . . . . . . . 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 6889	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘(𝑥 ∪ 𝑦)) = (𝐹‘𝑥))
27	22, 26	eqtr3d 2797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → ((𝐹‘𝑥) ∪ (𝐹‘𝑦)) = (𝐹‘𝑥))
28		ssequn2 4187	. . . . . . 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 43802	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
32		istopclsd.z	. . . . . 6 ⊢ (𝜑 → (𝐹‘∅) = ∅)
33		0elpw 5316	. . . . . . 7 ⊢ ∅ ∈ 𝒫 𝐵
34		fnelfp 7180	. . . . . . 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 4133	. . . . . . . . . . . . 13 ⊢ (𝐹 ∩ I ) ⊆ 𝐹
39		dmss 5883	. . . . . . . . . . . . 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 6729	. . . . . . . . . . 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 3909	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ∈ 𝒫 𝐵)
45	44	elpwid 4565	. . . . . . . 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 3909	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ∈ 𝒫 𝐵)
48	47	elpwid 4565	. . . . . . . 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 7180	. . . . . . . . . 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 7180	. . . . . . . . . 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 4168	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → ((𝐹‘𝑥) ∪ (𝐹‘𝑦)) = (𝑥 ∪ 𝑦))
58	49, 57	eqtrd 2795	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘(𝑥 ∪ 𝑦)) = (𝑥 ∪ 𝑦))
59	45, 48	unssd 4190	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥 ∪ 𝑦) ⊆ 𝐵)
60		vex 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
61		vex 3454	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
62	60, 61	unex 7761	. . . . . . . . 9 ⊢ (𝑥 ∪ 𝑦) ∈ V
63	62	elpw 4560	. . . . . . . 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 7180	. . . . . . 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 2760	. . . . 5 ⊢ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )}
69	31, 36, 67, 68	mretopd 23415	. . . 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 2860	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵))
72		topontop 23236	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
73	71, 72	syl 18	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
74		eqid 2760	. . . . . 6 ⊢ (mrCls‘(Clsd‘𝐽)) = (mrCls‘(Clsd‘𝐽))
75	74	mrccls 23402	. . . . 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 6889	. . . . . 6 ⊢ (𝜑 → (Clsd‘𝐽) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )}))
79	77, 78	eqtr4d 2798	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ I ) = (Clsd‘𝐽))
80	79	fveq2d 6889	. . . 4 ⊢ (𝜑 → (mrCls‘dom (𝐹 ∩ I )) = (mrCls‘(Clsd‘𝐽)))
81	76, 80	eqtr4d 2798	. . 3 ⊢ (𝜑 → (cls‘𝐽) = (mrCls‘dom (𝐹 ∩ I )))
82	6, 2, 16, 29, 30	ismrcd2 43803	. . 3 ⊢ (𝜑 → 𝐹 = (mrCls‘dom (𝐹 ∩ I )))
83	81, 82	eqtr4d 2798	. 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 1569   ∈ wcel 2144  {crab 3412   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  𝒫 cpw 4556   I cid 5544  dom cdm 5650   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  mrClscmrc 17758  Topctop 23216  TopOnctopon 23233  Clsdccld 23339  clsccl 23341

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-mre 17761  df-mrc 17762  df-top 23217  df-topon 23234  df-cld 23342  df-cls 23344

This theorem is used by: (None)