Theorem istopclsd 40522

Description: A closure function which satisfies sscls 22207, clsidm 22218, cls0 22231, and clsun 34517 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 6601	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝒫 𝐵)
4	3	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → 𝐹 Fn 𝒫 𝐵)
5		difss 4066	. . . . . . . . 9 ⊢ (𝐵 ∖ 𝑧) ⊆ 𝐵
6		istopclsd.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		elpw2g 5268	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑉 → ((𝐵 ∖ 𝑧) ∈ 𝒫 𝐵 ↔ (𝐵 ∖ 𝑧) ⊆ 𝐵))
8	6, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝐵 ∖ 𝑧) ∈ 𝒫 𝐵 ↔ (𝐵 ∖ 𝑧) ⊆ 𝐵))
9	5, 8	mpbiri 257	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∖ 𝑧) ∈ 𝒫 𝐵)
10	9	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑧) ∈ 𝒫 𝐵)
11		fnelfp 7047	. . . . . . 7 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ (𝐵 ∖ 𝑧) ∈ 𝒫 𝐵) → ((𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)))
12	4, 10, 11	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → ((𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)))
13	12	bicomd 222	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → ((𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧) ↔ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )))
14	13	rabbidva 3413	. . . 4 ⊢ (𝜑 → {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )})
15	1, 14	eqtrid 2790	. . 3 ⊢ (𝜑 → 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )})
16		istopclsd.e	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
17		simp1 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → 𝜑)
18		simp2 1136	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → 𝑥 ⊆ 𝐵)
19		simp3 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → 𝑦 ⊆ 𝑥)
20	19, 18	sstrd 3931	. . . . . . . . 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 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘(𝑥 ∪ 𝑦)) = ((𝐹‘𝑥) ∪ (𝐹‘𝑦)))
23		ssequn2 4117	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑥 ↔ (𝑥 ∪ 𝑦) = 𝑥)
24	23	biimpi 215	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ∪ 𝑦) = 𝑥)
25	24	3ad2ant3 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∪ 𝑦) = 𝑥)
26	25	fveq2d 6778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘(𝑥 ∪ 𝑦)) = (𝐹‘𝑥))
27	22, 26	eqtr3d 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → ((𝐹‘𝑥) ∪ (𝐹‘𝑦)) = (𝐹‘𝑥))
28		ssequn2 4117	. . . . . . 7 ⊢ ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ ((𝐹‘𝑥) ∪ (𝐹‘𝑦)) = (𝐹‘𝑥))
29	27, 28	sylibr 233	. . . . . 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 40520	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
32		istopclsd.z	. . . . . 6 ⊢ (𝜑 → (𝐹‘∅) = ∅)
33		0elpw 5278	. . . . . . 7 ⊢ ∅ ∈ 𝒫 𝐵
34		fnelfp 7047	. . . . . . 7 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) → (∅ ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∅) = ∅))
35	3, 33, 34	sylancl 586	. . . . . 6 ⊢ (𝜑 → (∅ ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∅) = ∅))
36	32, 35	mpbird 256	. . . . 5 ⊢ (𝜑 → ∅ ∈ dom (𝐹 ∩ I ))
37		simp1 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝜑)
38		inss1 4162	. . . . . . . . . . . . 13 ⊢ (𝐹 ∩ I ) ⊆ 𝐹
39		dmss 5811	. . . . . . . . . . . . 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 6620	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
42	41	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
43		simp2 1136	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ∈ dom (𝐹 ∩ I ))
44	42, 43	sseldd 3922	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ∈ 𝒫 𝐵)
45	44	elpwid 4544	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ⊆ 𝐵)
46		simp3 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ∈ dom (𝐹 ∩ I ))
47	42, 46	sseldd 3922	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ∈ 𝒫 𝐵)
48	47	elpwid 4544	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ⊆ 𝐵)
49	37, 45, 48, 21	syl3anc 1370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘(𝑥 ∪ 𝑦)) = ((𝐹‘𝑥) ∪ (𝐹‘𝑦)))
50	3	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝐹 Fn 𝒫 𝐵)
51		fnelfp 7047	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝑥 ∈ 𝒫 𝐵) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
52	50, 44, 51	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
53	43, 52	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘𝑥) = 𝑥)
54		fnelfp 7047	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝑦 ∈ 𝒫 𝐵) → (𝑦 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑦) = 𝑦))
55	50, 47, 54	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑦 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑦) = 𝑦))
56	46, 55	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘𝑦) = 𝑦)
57	53, 56	uneq12d 4098	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → ((𝐹‘𝑥) ∪ (𝐹‘𝑦)) = (𝑥 ∪ 𝑦))
58	49, 57	eqtrd 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘(𝑥 ∪ 𝑦)) = (𝑥 ∪ 𝑦))
59	45, 48	unssd 4120	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥 ∪ 𝑦) ⊆ 𝐵)
60		vex 3436	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
61		vex 3436	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
62	60, 61	unex 7596	. . . . . . . . 9 ⊢ (𝑥 ∪ 𝑦) ∈ V
63	62	elpw 4537	. . . . . . . 8 ⊢ ((𝑥 ∪ 𝑦) ∈ 𝒫 𝐵 ↔ (𝑥 ∪ 𝑦) ⊆ 𝐵)
64	59, 63	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥 ∪ 𝑦) ∈ 𝒫 𝐵)
65		fnelfp 7047	. . . . . . 7 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ (𝑥 ∪ 𝑦) ∈ 𝒫 𝐵) → ((𝑥 ∪ 𝑦) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝑥 ∪ 𝑦)) = (𝑥 ∪ 𝑦)))
66	50, 64, 65	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → ((𝑥 ∪ 𝑦) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝑥 ∪ 𝑦)) = (𝑥 ∪ 𝑦)))
67	58, 66	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥 ∪ 𝑦) ∈ dom (𝐹 ∩ I ))
68		eqid 2738	. . . . 5 ⊢ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )}
69	31, 36, 67, 68	mretopd 22243	. . . 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 495	. . 3 ⊢ (𝜑 → {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )} ∈ (TopOn‘𝐵))
71	15, 70	eqeltrd 2839	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵))
72		topontop 22062	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
73	71, 72	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
74		eqid 2738	. . . . . 6 ⊢ (mrCls‘(Clsd‘𝐽)) = (mrCls‘(Clsd‘𝐽))
75	74	mrccls 22230	. . . . 5 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (mrCls‘(Clsd‘𝐽)))
76	73, 75	syl 17	. . . 4 ⊢ (𝜑 → (cls‘𝐽) = (mrCls‘(Clsd‘𝐽)))
77	69	simprd 496	. . . . . 6 ⊢ (𝜑 → dom (𝐹 ∩ I ) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )}))
78	15	fveq2d 6778	. . . . . 6 ⊢ (𝜑 → (Clsd‘𝐽) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )}))
79	77, 78	eqtr4d 2781	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ I ) = (Clsd‘𝐽))
80	79	fveq2d 6778	. . . 4 ⊢ (𝜑 → (mrCls‘dom (𝐹 ∩ I )) = (mrCls‘(Clsd‘𝐽)))
81	76, 80	eqtr4d 2781	. . 3 ⊢ (𝜑 → (cls‘𝐽) = (mrCls‘dom (𝐹 ∩ I )))
82	6, 2, 16, 29, 30	ismrcd2 40521	. . 3 ⊢ (𝜑 → 𝐹 = (mrCls‘dom (𝐹 ∩ I )))
83	81, 82	eqtr4d 2781	. 2 ⊢ (𝜑 → (cls‘𝐽) = 𝐹)
84	71, 83	jca 512	1 ⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {crab 3068   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533   I cid 5488  dom cdm 5589   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  mrClscmrc 17292  Topctop 22042  TopOnctopon 22059  Clsdccld 22167  clsccl 22169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-mre 17295  df-mrc 17296  df-top 22043  df-topon 22060  df-cld 22170  df-cls 22172

This theorem is referenced by: (None)