Theorem istopclsd 40225

Description: A closure function which satisfies sscls 21953, clsidm 21964, cls0 21977, and clsun 34254 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 6546	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝒫 𝐵)
4	3	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → 𝐹 Fn 𝒫 𝐵)
5		difss 4046	. . . . . . . . 9 ⊢ (𝐵 ∖ 𝑧) ⊆ 𝐵
6		istopclsd.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		elpw2g 5237	. . . . . . . . . 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 261	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∖ 𝑧) ∈ 𝒫 𝐵)
10	9	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑧) ∈ 𝒫 𝐵)
11		fnelfp 6990	. . . . . . 7 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ (𝐵 ∖ 𝑧) ∈ 𝒫 𝐵) → ((𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)))
12	4, 10, 11	syl2anc 587	. . . . . 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 3388	. . . 4 ⊢ (𝜑 → {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵 ∖ 𝑧)) = (𝐵 ∖ 𝑧)} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )})
15	1, 14	syl5eq 2790	. . 3 ⊢ (𝜑 → 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )})
16		istopclsd.e	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
17		simp1 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → 𝜑)
18		simp2 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → 𝑥 ⊆ 𝐵)
19		simp3 1140	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → 𝑦 ⊆ 𝑥)
20	19, 18	sstrd 3911	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘(𝑥 ∪ 𝑦)) = ((𝐹‘𝑥) ∪ (𝐹‘𝑦)))
23		ssequn2 4097	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝑥 ↔ (𝑥 ∪ 𝑦) = 𝑥)
24	23	biimpi 219	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ∪ 𝑦) = 𝑥)
25	24	3ad2ant3 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∪ 𝑦) = 𝑥)
26	25	fveq2d 6721	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘(𝑥 ∪ 𝑦)) = (𝐹‘𝑥))
27	22, 26	eqtr3d 2779	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → ((𝐹‘𝑥) ∪ (𝐹‘𝑦)) = (𝐹‘𝑥))
28		ssequn2 4097	. . . . . . 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 40223	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
32		istopclsd.z	. . . . . 6 ⊢ (𝜑 → (𝐹‘∅) = ∅)
33		0elpw 5247	. . . . . . 7 ⊢ ∅ ∈ 𝒫 𝐵
34		fnelfp 6990	. . . . . . 7 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) → (∅ ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∅) = ∅))
35	3, 33, 34	sylancl 589	. . . . . 6 ⊢ (𝜑 → (∅ ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∅) = ∅))
36	32, 35	mpbird 260	. . . . 5 ⊢ (𝜑 → ∅ ∈ dom (𝐹 ∩ I ))
37		simp1 1138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝜑)
38		inss1 4143	. . . . . . . . . . . . 13 ⊢ (𝐹 ∩ I ) ⊆ 𝐹
39		dmss 5771	. . . . . . . . . . . . 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 6565	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
42	41	3ad2ant1 1135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
43		simp2 1139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ∈ dom (𝐹 ∩ I ))
44	42, 43	sseldd 3902	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ∈ 𝒫 𝐵)
45	44	elpwid 4524	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ⊆ 𝐵)
46		simp3 1140	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ∈ dom (𝐹 ∩ I ))
47	42, 46	sseldd 3902	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ∈ 𝒫 𝐵)
48	47	elpwid 4524	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ⊆ 𝐵)
49	37, 45, 48, 21	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘(𝑥 ∪ 𝑦)) = ((𝐹‘𝑥) ∪ (𝐹‘𝑦)))
50	3	3ad2ant1 1135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝐹 Fn 𝒫 𝐵)
51		fnelfp 6990	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝑥 ∈ 𝒫 𝐵) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
52	50, 44, 51	syl2anc 587	. . . . . . . . 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 6990	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝑦 ∈ 𝒫 𝐵) → (𝑦 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑦) = 𝑦))
55	50, 47, 54	syl2anc 587	. . . . . . . . 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 4078	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → ((𝐹‘𝑥) ∪ (𝐹‘𝑦)) = (𝑥 ∪ 𝑦))
58	49, 57	eqtrd 2777	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘(𝑥 ∪ 𝑦)) = (𝑥 ∪ 𝑦))
59	45, 48	unssd 4100	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥 ∪ 𝑦) ⊆ 𝐵)
60		vex 3412	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
61		vex 3412	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
62	60, 61	unex 7531	. . . . . . . . 9 ⊢ (𝑥 ∪ 𝑦) ∈ V
63	62	elpw 4517	. . . . . . . 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 6990	. . . . . . 7 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ (𝑥 ∪ 𝑦) ∈ 𝒫 𝐵) → ((𝑥 ∪ 𝑦) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝑥 ∪ 𝑦)) = (𝑥 ∪ 𝑦)))
66	50, 64, 65	syl2anc 587	. . . . . 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 2737	. . . . 5 ⊢ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )}
69	31, 36, 67, 68	mretopd 21989	. . . 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 498	. . 3 ⊢ (𝜑 → {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )} ∈ (TopOn‘𝐵))
71	15, 70	eqeltrd 2838	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵))
72		topontop 21810	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
73	71, 72	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
74		eqid 2737	. . . . . 6 ⊢ (mrCls‘(Clsd‘𝐽)) = (mrCls‘(Clsd‘𝐽))
75	74	mrccls 21976	. . . . 5 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (mrCls‘(Clsd‘𝐽)))
76	73, 75	syl 17	. . . 4 ⊢ (𝜑 → (cls‘𝐽) = (mrCls‘(Clsd‘𝐽)))
77	69	simprd 499	. . . . . 6 ⊢ (𝜑 → dom (𝐹 ∩ I ) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )}))
78	15	fveq2d 6721	. . . . . 6 ⊢ (𝜑 → (Clsd‘𝐽) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑧) ∈ dom (𝐹 ∩ I )}))
79	77, 78	eqtr4d 2780	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ I ) = (Clsd‘𝐽))
80	79	fveq2d 6721	. . . 4 ⊢ (𝜑 → (mrCls‘dom (𝐹 ∩ I )) = (mrCls‘(Clsd‘𝐽)))
81	76, 80	eqtr4d 2780	. . 3 ⊢ (𝜑 → (cls‘𝐽) = (mrCls‘dom (𝐹 ∩ I )))
82	6, 2, 16, 29, 30	ismrcd2 40224	. . 3 ⊢ (𝜑 → 𝐹 = (mrCls‘dom (𝐹 ∩ I )))
83	81, 82	eqtr4d 2780	. 2 ⊢ (𝜑 → (cls‘𝐽) = 𝐹)
84	71, 83	jca 515	1 ⊢ (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  {crab 3065   ∖ cdif 3863   ∪ cun 3864   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  𝒫 cpw 4513   I cid 5454  dom cdm 5551   Fn wfn 6375  ⟶wf 6376  ‘cfv 6380  mrClscmrc 17086  Topctop 21790  TopOnctopon 21807  Clsdccld 21913  clsccl 21915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-mre 17089  df-mrc 17090  df-top 21791  df-topon 21808  df-cld 21916  df-cls 21918

This theorem is referenced by: (None)