Metamath Proof Explorer


Theorem ismrcd1

Description: Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid ), isotone (satisfies mrcss ), and idempotent (satisfies mrcidm ) 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 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015)

Ref Expression
Hypotheses ismrcd.b ( 𝜑𝐵𝑉 )
ismrcd.f ( 𝜑𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 )
ismrcd.e ( ( 𝜑𝑥𝐵 ) → 𝑥 ⊆ ( 𝐹𝑥 ) )
ismrcd.m ( ( 𝜑𝑥𝐵𝑦𝑥 ) → ( 𝐹𝑦 ) ⊆ ( 𝐹𝑥 ) )
ismrcd.i ( ( 𝜑𝑥𝐵 ) → ( 𝐹 ‘ ( 𝐹𝑥 ) ) = ( 𝐹𝑥 ) )
Assertion ismrcd1 ( 𝜑 → dom ( 𝐹 ∩ I ) ∈ ( Moore ‘ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ismrcd.b ( 𝜑𝐵𝑉 )
2 ismrcd.f ( 𝜑𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 )
3 ismrcd.e ( ( 𝜑𝑥𝐵 ) → 𝑥 ⊆ ( 𝐹𝑥 ) )
4 ismrcd.m ( ( 𝜑𝑥𝐵𝑦𝑥 ) → ( 𝐹𝑦 ) ⊆ ( 𝐹𝑥 ) )
5 ismrcd.i ( ( 𝜑𝑥𝐵 ) → ( 𝐹 ‘ ( 𝐹𝑥 ) ) = ( 𝐹𝑥 ) )
6 inss1 ( 𝐹 ∩ I ) ⊆ 𝐹
7 dmss ( ( 𝐹 ∩ I ) ⊆ 𝐹 → dom ( 𝐹 ∩ I ) ⊆ dom 𝐹 )
8 6 7 ax-mp dom ( 𝐹 ∩ I ) ⊆ dom 𝐹
9 8 2 fssdm ( 𝜑 → dom ( 𝐹 ∩ I ) ⊆ 𝒫 𝐵 )
10 ssid 𝐵𝐵
11 elpwg ( 𝐵𝑉 → ( 𝐵 ∈ 𝒫 𝐵𝐵𝐵 ) )
12 1 11 syl ( 𝜑 → ( 𝐵 ∈ 𝒫 𝐵𝐵𝐵 ) )
13 10 12 mpbiri ( 𝜑𝐵 ∈ 𝒫 𝐵 )
14 2 13 ffvelrnd ( 𝜑 → ( 𝐹𝐵 ) ∈ 𝒫 𝐵 )
15 14 elpwid ( 𝜑 → ( 𝐹𝐵 ) ⊆ 𝐵 )
16 velpw ( 𝑥 ∈ 𝒫 𝐵𝑥𝐵 )
17 16 3 sylan2b ( ( 𝜑𝑥 ∈ 𝒫 𝐵 ) → 𝑥 ⊆ ( 𝐹𝑥 ) )
18 17 ralrimiva ( 𝜑 → ∀ 𝑥 ∈ 𝒫 𝐵 𝑥 ⊆ ( 𝐹𝑥 ) )
19 id ( 𝑥 = 𝐵𝑥 = 𝐵 )
20 fveq2 ( 𝑥 = 𝐵 → ( 𝐹𝑥 ) = ( 𝐹𝐵 ) )
21 19 20 sseq12d ( 𝑥 = 𝐵 → ( 𝑥 ⊆ ( 𝐹𝑥 ) ↔ 𝐵 ⊆ ( 𝐹𝐵 ) ) )
22 21 rspcva ( ( 𝐵 ∈ 𝒫 𝐵 ∧ ∀ 𝑥 ∈ 𝒫 𝐵 𝑥 ⊆ ( 𝐹𝑥 ) ) → 𝐵 ⊆ ( 𝐹𝐵 ) )
23 13 18 22 syl2anc ( 𝜑𝐵 ⊆ ( 𝐹𝐵 ) )
24 15 23 eqssd ( 𝜑 → ( 𝐹𝐵 ) = 𝐵 )
25 2 ffnd ( 𝜑𝐹 Fn 𝒫 𝐵 )
26 fnelfp ( ( 𝐹 Fn 𝒫 𝐵𝐵 ∈ 𝒫 𝐵 ) → ( 𝐵 ∈ dom ( 𝐹 ∩ I ) ↔ ( 𝐹𝐵 ) = 𝐵 ) )
27 25 13 26 syl2anc ( 𝜑 → ( 𝐵 ∈ dom ( 𝐹 ∩ I ) ↔ ( 𝐹𝐵 ) = 𝐵 ) )
28 24 27 mpbird ( 𝜑𝐵 ∈ dom ( 𝐹 ∩ I ) )
29 simp2 ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → 𝑧 ⊆ dom ( 𝐹 ∩ I ) )
30 9 3ad2ant1 ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → dom ( 𝐹 ∩ I ) ⊆ 𝒫 𝐵 )
31 29 30 sstrd ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → 𝑧 ⊆ 𝒫 𝐵 )
32 simp3 ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → 𝑧 ≠ ∅ )
33 intssuni2 ( ( 𝑧 ⊆ 𝒫 𝐵𝑧 ≠ ∅ ) → 𝑧 𝒫 𝐵 )
34 31 32 33 syl2anc ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → 𝑧 𝒫 𝐵 )
35 unipw 𝒫 𝐵 = 𝐵
36 34 35 sseqtrdi ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → 𝑧𝐵 )
37 intex ( 𝑧 ≠ ∅ ↔ 𝑧 ∈ V )
38 elpwg ( 𝑧 ∈ V → ( 𝑧 ∈ 𝒫 𝐵 𝑧𝐵 ) )
39 37 38 sylbi ( 𝑧 ≠ ∅ → ( 𝑧 ∈ 𝒫 𝐵 𝑧𝐵 ) )
40 39 3ad2ant3 ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ( 𝑧 ∈ 𝒫 𝐵 𝑧𝐵 ) )
41 36 40 mpbird ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → 𝑧 ∈ 𝒫 𝐵 )
42 41 adantr ( ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥𝑧 ) → 𝑧 ∈ 𝒫 𝐵 )
43 4 3expib ( 𝜑 → ( ( 𝑥𝐵𝑦𝑥 ) → ( 𝐹𝑦 ) ⊆ ( 𝐹𝑥 ) ) )
44 43 alrimiv ( 𝜑 → ∀ 𝑦 ( ( 𝑥𝐵𝑦𝑥 ) → ( 𝐹𝑦 ) ⊆ ( 𝐹𝑥 ) ) )
45 44 3ad2ant1 ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ∀ 𝑦 ( ( 𝑥𝐵𝑦𝑥 ) → ( 𝐹𝑦 ) ⊆ ( 𝐹𝑥 ) ) )
46 45 adantr ( ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥𝑧 ) → ∀ 𝑦 ( ( 𝑥𝐵𝑦𝑥 ) → ( 𝐹𝑦 ) ⊆ ( 𝐹𝑥 ) ) )
47 31 sselda ( ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥𝑧 ) → 𝑥 ∈ 𝒫 𝐵 )
48 47 elpwid ( ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥𝑧 ) → 𝑥𝐵 )
49 intss1 ( 𝑥𝑧 𝑧𝑥 )
50 49 adantl ( ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥𝑧 ) → 𝑧𝑥 )
51 48 50 jca ( ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥𝑧 ) → ( 𝑥𝐵 𝑧𝑥 ) )
52 sseq1 ( 𝑦 = 𝑧 → ( 𝑦𝑥 𝑧𝑥 ) )
53 52 anbi2d ( 𝑦 = 𝑧 → ( ( 𝑥𝐵𝑦𝑥 ) ↔ ( 𝑥𝐵 𝑧𝑥 ) ) )
54 fveq2 ( 𝑦 = 𝑧 → ( 𝐹𝑦 ) = ( 𝐹 𝑧 ) )
55 54 sseq1d ( 𝑦 = 𝑧 → ( ( 𝐹𝑦 ) ⊆ ( 𝐹𝑥 ) ↔ ( 𝐹 𝑧 ) ⊆ ( 𝐹𝑥 ) ) )
56 53 55 imbi12d ( 𝑦 = 𝑧 → ( ( ( 𝑥𝐵𝑦𝑥 ) → ( 𝐹𝑦 ) ⊆ ( 𝐹𝑥 ) ) ↔ ( ( 𝑥𝐵 𝑧𝑥 ) → ( 𝐹 𝑧 ) ⊆ ( 𝐹𝑥 ) ) ) )
57 56 spcgv ( 𝑧 ∈ 𝒫 𝐵 → ( ∀ 𝑦 ( ( 𝑥𝐵𝑦𝑥 ) → ( 𝐹𝑦 ) ⊆ ( 𝐹𝑥 ) ) → ( ( 𝑥𝐵 𝑧𝑥 ) → ( 𝐹 𝑧 ) ⊆ ( 𝐹𝑥 ) ) ) )
58 42 46 51 57 syl3c ( ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥𝑧 ) → ( 𝐹 𝑧 ) ⊆ ( 𝐹𝑥 ) )
59 29 sselda ( ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥𝑧 ) → 𝑥 ∈ dom ( 𝐹 ∩ I ) )
60 25 3ad2ant1 ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → 𝐹 Fn 𝒫 𝐵 )
61 60 adantr ( ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥𝑧 ) → 𝐹 Fn 𝒫 𝐵 )
62 fnelfp ( ( 𝐹 Fn 𝒫 𝐵𝑥 ∈ 𝒫 𝐵 ) → ( 𝑥 ∈ dom ( 𝐹 ∩ I ) ↔ ( 𝐹𝑥 ) = 𝑥 ) )
63 61 47 62 syl2anc ( ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥𝑧 ) → ( 𝑥 ∈ dom ( 𝐹 ∩ I ) ↔ ( 𝐹𝑥 ) = 𝑥 ) )
64 59 63 mpbid ( ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥𝑧 ) → ( 𝐹𝑥 ) = 𝑥 )
65 58 64 sseqtrd ( ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) ∧ 𝑥𝑧 ) → ( 𝐹 𝑧 ) ⊆ 𝑥 )
66 65 ralrimiva ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ∀ 𝑥𝑧 ( 𝐹 𝑧 ) ⊆ 𝑥 )
67 ssint ( ( 𝐹 𝑧 ) ⊆ 𝑧 ↔ ∀ 𝑥𝑧 ( 𝐹 𝑧 ) ⊆ 𝑥 )
68 66 67 sylibr ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ( 𝐹 𝑧 ) ⊆ 𝑧 )
69 18 3ad2ant1 ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ∀ 𝑥 ∈ 𝒫 𝐵 𝑥 ⊆ ( 𝐹𝑥 ) )
70 id ( 𝑥 = 𝑧𝑥 = 𝑧 )
71 fveq2 ( 𝑥 = 𝑧 → ( 𝐹𝑥 ) = ( 𝐹 𝑧 ) )
72 70 71 sseq12d ( 𝑥 = 𝑧 → ( 𝑥 ⊆ ( 𝐹𝑥 ) ↔ 𝑧 ⊆ ( 𝐹 𝑧 ) ) )
73 72 rspcva ( ( 𝑧 ∈ 𝒫 𝐵 ∧ ∀ 𝑥 ∈ 𝒫 𝐵 𝑥 ⊆ ( 𝐹𝑥 ) ) → 𝑧 ⊆ ( 𝐹 𝑧 ) )
74 41 69 73 syl2anc ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → 𝑧 ⊆ ( 𝐹 𝑧 ) )
75 68 74 eqssd ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ( 𝐹 𝑧 ) = 𝑧 )
76 fnelfp ( ( 𝐹 Fn 𝒫 𝐵 𝑧 ∈ 𝒫 𝐵 ) → ( 𝑧 ∈ dom ( 𝐹 ∩ I ) ↔ ( 𝐹 𝑧 ) = 𝑧 ) )
77 60 41 76 syl2anc ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → ( 𝑧 ∈ dom ( 𝐹 ∩ I ) ↔ ( 𝐹 𝑧 ) = 𝑧 ) )
78 75 77 mpbird ( ( 𝜑𝑧 ⊆ dom ( 𝐹 ∩ I ) ∧ 𝑧 ≠ ∅ ) → 𝑧 ∈ dom ( 𝐹 ∩ I ) )
79 9 28 78 ismred ( 𝜑 → dom ( 𝐹 ∩ I ) ∈ ( Moore ‘ 𝐵 ) )