Theorem ismrcd1 43147

Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 17577), isotone (satisfies mrcss 17576), and idempotent (satisfies mrcidm 17579) has a collection of fixed points which is a Moore collection, and itself is the closure operator for that collection. This can be taken as an alternate definition for the closure operators. This is the first half, ismrcd2 43148 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses

Ref	Expression
ismrcd.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
ismrcd.f	⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
ismrcd.e	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
ismrcd.m	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥))
ismrcd.i	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → (𝐹‘(𝐹‘𝑥)) = (𝐹‘𝑥))

Assertion

Ref	Expression
ismrcd1	⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))

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

Proof of Theorem ismrcd1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 4178	. . . 4 ⊢ (𝐹 ∩ I ) ⊆ 𝐹
2		dmss 5852	. . . 4 ⊢ ((𝐹 ∩ I ) ⊆ 𝐹 → dom (𝐹 ∩ I ) ⊆ dom 𝐹)
3	1, 2	ax-mp 5	. . 3 ⊢ dom (𝐹 ∩ I ) ⊆ dom 𝐹
4		ismrcd.f	. . 3 ⊢ (𝜑 → 𝐹:𝒫 𝐵⟶𝒫 𝐵)
5	3, 4	fssdm 6682	. 2 ⊢ (𝜑 → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
6		ssid 3945	. . . . . . 7 ⊢ 𝐵 ⊆ 𝐵
7		ismrcd.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
8		elpwg 4545	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ 𝒫 𝐵 ↔ 𝐵 ⊆ 𝐵))
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ 𝒫 𝐵 ↔ 𝐵 ⊆ 𝐵))
10	6, 9	mpbiri 258	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵)
11	4, 10	ffvelcdmd 7032	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ 𝒫 𝐵)
12	11	elpwid 4551	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ⊆ 𝐵)
13		velpw 4547	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
14		ismrcd.e	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
15	13, 14	sylan2b 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐵) → 𝑥 ⊆ (𝐹‘𝑥))
16	15	ralrimiva 3130	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥))
17		id 22	. . . . . . 7 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
18		fveq2 6835	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
19	17, 18	sseq12d 3956	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝐵 ⊆ (𝐹‘𝐵)))
20	19	rspcva 3563	. . . . 5 ⊢ ((𝐵 ∈ 𝒫 𝐵 ∧ ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥)) → 𝐵 ⊆ (𝐹‘𝐵))
21	10, 16, 20	syl2anc 585	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐹‘𝐵))
22	12, 21	eqssd 3940	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) = 𝐵)
23	4	ffnd 6664	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝒫 𝐵)
24		fnelfp 7124	. . . 4 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝐵 ∈ 𝒫 𝐵) → (𝐵 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝐵) = 𝐵))
25	23, 10, 24	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐵 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝐵) = 𝐵))
26	22, 25	mpbird 257	. 2 ⊢ (𝜑 → 𝐵 ∈ dom (𝐹 ∩ I ))
27		simp2 1138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝑧 ⊆ dom (𝐹 ∩ I ))
28	5	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
29	27, 28	sstrd 3933	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝑧 ⊆ 𝒫 𝐵)
30		simp3 1139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝑧 ≠ ∅)
31		intssuni2 4916	. . . . . . . . . . . 12 ⊢ ((𝑧 ⊆ 𝒫 𝐵 ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ ∪ 𝒫 𝐵)
32	29, 30, 31	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ ∪ 𝒫 𝐵)
33		unipw 5398	. . . . . . . . . . 11 ⊢ ∪ 𝒫 𝐵 = 𝐵
34	32, 33	sseqtrdi 3963	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ 𝐵)
35		intex 5282	. . . . . . . . . . . 12 ⊢ (𝑧 ≠ ∅ ↔ ∩ 𝑧 ∈ V)
36		elpwg 4545	. . . . . . . . . . . 12 ⊢ (∩ 𝑧 ∈ V → (∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵))
37	35, 36	sylbi 217	. . . . . . . . . . 11 ⊢ (𝑧 ≠ ∅ → (∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵))
38	37	3ad2ant3 1136	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (∩ 𝑧 ∈ 𝒫 𝐵 ↔ ∩ 𝑧 ⊆ 𝐵))
39	34, 38	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ 𝒫 𝐵)
40	39	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → ∩ 𝑧 ∈ 𝒫 𝐵)
41		ismrcd.m	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥))
42	41	3expib 1123	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
43	42	alrimiv 1929	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
44	43	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
45	44	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → ∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)))
46	29	sselda 3922	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝒫 𝐵)
47	46	elpwid 4551	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ⊆ 𝐵)
48		intss1 4906	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑧 → ∩ 𝑧 ⊆ 𝑥)
49	48	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → ∩ 𝑧 ⊆ 𝑥)
50	47, 49	jca 511	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥))
51		sseq1 3948	. . . . . . . . . . 11 ⊢ (𝑦 = ∩ 𝑧 → (𝑦 ⊆ 𝑥 ↔ ∩ 𝑧 ⊆ 𝑥))
52	51	anbi2d 631	. . . . . . . . . 10 ⊢ (𝑦 = ∩ 𝑧 → ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) ↔ (𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥)))
53		fveq2 6835	. . . . . . . . . . 11 ⊢ (𝑦 = ∩ 𝑧 → (𝐹‘𝑦) = (𝐹‘∩ 𝑧))
54	53	sseq1d 3954	. . . . . . . . . 10 ⊢ (𝑦 = ∩ 𝑧 → ((𝐹‘𝑦) ⊆ (𝐹‘𝑥) ↔ (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥)))
55	52, 54	imbi12d 344	. . . . . . . . 9 ⊢ (𝑦 = ∩ 𝑧 → (((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) ↔ ((𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥) → (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥))))
56	55	spcgv 3539	. . . . . . . 8 ⊢ (∩ 𝑧 ∈ 𝒫 𝐵 → (∀𝑦((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥) → (𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → ((𝑥 ⊆ 𝐵 ∧ ∩ 𝑧 ⊆ 𝑥) → (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥))))
57	40, 45, 50, 56	syl3c 66	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝐹‘∩ 𝑧) ⊆ (𝐹‘𝑥))
58	27	sselda 3922	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ dom (𝐹 ∩ I ))
59	23	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → 𝐹 Fn 𝒫 𝐵)
60	59	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → 𝐹 Fn 𝒫 𝐵)
61		fnelfp 7124	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ 𝑥 ∈ 𝒫 𝐵) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
62	60, 46, 61	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
63	58, 62	mpbid 232	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝐹‘𝑥) = 𝑥)
64	57, 63	sseqtrd 3959	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) ∧ 𝑥 ∈ 𝑧) → (𝐹‘∩ 𝑧) ⊆ 𝑥)
65	64	ralrimiva 3130	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∀𝑥 ∈ 𝑧 (𝐹‘∩ 𝑧) ⊆ 𝑥)
66		ssint 4907	. . . . 5 ⊢ ((𝐹‘∩ 𝑧) ⊆ ∩ 𝑧 ↔ ∀𝑥 ∈ 𝑧 (𝐹‘∩ 𝑧) ⊆ 𝑥)
67	65, 66	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (𝐹‘∩ 𝑧) ⊆ ∩ 𝑧)
68	16	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥))
69		id 22	. . . . . . 7 ⊢ (𝑥 = ∩ 𝑧 → 𝑥 = ∩ 𝑧)
70		fveq2 6835	. . . . . . 7 ⊢ (𝑥 = ∩ 𝑧 → (𝐹‘𝑥) = (𝐹‘∩ 𝑧))
71	69, 70	sseq12d 3956	. . . . . 6 ⊢ (𝑥 = ∩ 𝑧 → (𝑥 ⊆ (𝐹‘𝑥) ↔ ∩ 𝑧 ⊆ (𝐹‘∩ 𝑧)))
72	71	rspcva 3563	. . . . 5 ⊢ ((∩ 𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑥 ∈ 𝒫 𝐵𝑥 ⊆ (𝐹‘𝑥)) → ∩ 𝑧 ⊆ (𝐹‘∩ 𝑧))
73	39, 68, 72	syl2anc 585	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ⊆ (𝐹‘∩ 𝑧))
74	67, 73	eqssd 3940	. . 3 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (𝐹‘∩ 𝑧) = ∩ 𝑧)
75		fnelfp 7124	. . . 4 ⊢ ((𝐹 Fn 𝒫 𝐵 ∧ ∩ 𝑧 ∈ 𝒫 𝐵) → (∩ 𝑧 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∩ 𝑧) = ∩ 𝑧))
76	59, 39, 75	syl2anc 585	. . 3 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → (∩ 𝑧 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∩ 𝑧) = ∩ 𝑧))
77	74, 76	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝑧 ⊆ dom (𝐹 ∩ I ) ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ dom (𝐹 ∩ I ))
78	5, 26, 77	ismred 17558	1 ⊢ (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ∪ cuni 4851  ∩ cint 4890   I cid 5519  dom cdm 5625   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  Moorecmre 17538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-mre 17542

This theorem is referenced by:  ismrcd2  43148  istopclsd  43149  ismrc  43150