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Metamath Proof Explorer

Theorem dislly

Description: The discrete space ~P X is locally A if and only if every singleton space has property A . (Contributed by Mario Carneiro, 20-Mar-2015)

Ref Expression
Assertion dislly ( 𝑋 ∈ 𝑉 → ( 𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀ 𝑥 ∈ 𝑋 𝒫 { 𝑥 } ∈ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 simplr ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) → 𝒫 𝑋 ∈ Locally 𝐴 )
2 vex ⊢ 𝑥 ∈ V
3 2 snelpw ⊢ ( 𝑥 ∈ 𝑋 ↔ { 𝑥 } ∈ 𝒫 𝑋 )
4 3 bilani ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) → { 𝑥 } ∈ 𝒫 𝑋 )
5 vsnid ⊢ 𝑥 ∈ { 𝑥 }
6 5 a1i ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) → 𝑥 ∈ { 𝑥 } )
7 llyi ⊢ ( ( 𝒫 𝑋 ∈ Locally 𝐴 ∧ { 𝑥 } ∈ 𝒫 𝑋 ∧ 𝑥 ∈ { 𝑥 } ) → ∃ 𝑦 ∈ 𝒫 𝑋 ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) )
8 1 4 6 7 syl3anc ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) → ∃ 𝑦 ∈ 𝒫 𝑋 ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) )
9 simpr1 ⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) ) → 𝑦 ⊆ { 𝑥 } )
10 simpr2 ⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) ) → 𝑥 ∈ 𝑦 )
11 10 snssd ⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) ) → { 𝑥 } ⊆ 𝑦 )
12 9 11 eqssd ⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) ) → 𝑦 = { 𝑥 } )
13 12 oveq2d ⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) ) → ( 𝒫 𝑋 ↾t 𝑦 ) = ( 𝒫 𝑋 ↾t { 𝑥 } ) )
14 simplll ⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) ) → 𝑋 ∈ 𝑉 )
15 simplr ⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) ) → 𝑥 ∈ 𝑋 )
16 15 snssd ⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) ) → { 𝑥 } ⊆ 𝑋 )
17 restdis ⊢ ( ( 𝑋 ∈ 𝑉 ∧ { 𝑥 } ⊆ 𝑋 ) → ( 𝒫 𝑋 ↾t { 𝑥 } ) = 𝒫 { 𝑥 } )
18 14 16 17 syl2anc ⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) ) → ( 𝒫 𝑋 ↾t { 𝑥 } ) = 𝒫 { 𝑥 } )
19 13 18 eqtrd ⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) ) → ( 𝒫 𝑋 ↾t 𝑦 ) = 𝒫 { 𝑥 } )
20 simpr3 ⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) ) → ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 )
21 19 20 eqeltrrd ⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) ∧ ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) ) → 𝒫 { 𝑥 } ∈ 𝐴 )
22 21 ex ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) → ( ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) → 𝒫 { 𝑥 } ∈ 𝐴 ) )
23 22 rexlimdvw ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) → ( ∃ 𝑦 ∈ 𝒫 𝑋 ( 𝑦 ⊆ { 𝑥 } ∧ 𝑥 ∈ 𝑦 ∧ ( 𝒫 𝑋 ↾t 𝑦 ) ∈ 𝐴 ) → 𝒫 { 𝑥 } ∈ 𝐴 ) )
24 8 23 mpd ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) ∧ 𝑥 ∈ 𝑋 ) → 𝒫 { 𝑥 } ∈ 𝐴 )
25 24 ralrimiva ⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴 ) → ∀ 𝑥 ∈ 𝑋 𝒫 { 𝑥 } ∈ 𝐴 )
26 distop ⊢ ( 𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Top )
27 26 adantr ⊢ ( ( 𝑋 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 𝒫 { 𝑥 } ∈ 𝐴 ) → 𝒫 𝑋 ∈ Top )
28 elpwi ⊢ ( 𝑦 ∈ 𝒫 𝑋 → 𝑦 ⊆ 𝑋 )
29 28 adantl ⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) → 𝑦 ⊆ 𝑋 )
30 ssralv ⊢ ( 𝑦 ⊆ 𝑋 → ( ∀ 𝑥 ∈ 𝑋 𝒫 { 𝑥 } ∈ 𝐴 → ∀ 𝑥 ∈ 𝑦 𝒫 { 𝑥 } ∈ 𝐴 ) )
31 29 30 syl ⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 𝒫 { 𝑥 } ∈ 𝐴 → ∀ 𝑥 ∈ 𝑦 𝒫 { 𝑥 } ∈ 𝐴 ) )
32 simprl ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝒫 { 𝑥 } ∈ 𝐴 ) ) → 𝑥 ∈ 𝑦 )
33 32 snssd ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝒫 { 𝑥 } ∈ 𝐴 ) ) → { 𝑥 } ⊆ 𝑦 )
34 29 adantr ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝒫 { 𝑥 } ∈ 𝐴 ) ) → 𝑦 ⊆ 𝑋 )
35 33 34 sstrd ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝒫 { 𝑥 } ∈ 𝐴 ) ) → { 𝑥 } ⊆ 𝑋 )
36 vsnex ⊢ { 𝑥 } ∈ V
37 36 elpw ⊢ ( { 𝑥 } ∈ 𝒫 𝑋 ↔ { 𝑥 } ⊆ 𝑋 )
38 35 37 sylibr ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝒫 { 𝑥 } ∈ 𝐴 ) ) → { 𝑥 } ∈ 𝒫 𝑋 )
39 36 elpw ⊢ ( { 𝑥 } ∈ 𝒫 𝑦 ↔ { 𝑥 } ⊆ 𝑦 )
40 33 39 sylibr ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝒫 { 𝑥 } ∈ 𝐴 ) ) → { 𝑥 } ∈ 𝒫 𝑦 )
41 38 40 elind ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝒫 { 𝑥 } ∈ 𝐴 ) ) → { 𝑥 } ∈ ( 𝒫 𝑋 ∩ 𝒫 𝑦 ) )
42 snidg ⊢ ( 𝑥 ∈ 𝑦 → 𝑥 ∈ { 𝑥 } )
43 42 ad2antrl ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝒫 { 𝑥 } ∈ 𝐴 ) ) → 𝑥 ∈ { 𝑥 } )
44 simpll ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝒫 { 𝑥 } ∈ 𝐴 ) ) → 𝑋 ∈ 𝑉 )
45 44 35 17 syl2anc ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝒫 { 𝑥 } ∈ 𝐴 ) ) → ( 𝒫 𝑋 ↾t { 𝑥 } ) = 𝒫 { 𝑥 } )
46 simprr ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝒫 { 𝑥 } ∈ 𝐴 ) ) → 𝒫 { 𝑥 } ∈ 𝐴 )
47 45 46 eqeltrd ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝒫 { 𝑥 } ∈ 𝐴 ) ) → ( 𝒫 𝑋 ↾t { 𝑥 } ) ∈ 𝐴 )
48 eleq2 ⊢ ( 𝑢 = { 𝑥 } → ( 𝑥 ∈ 𝑢 ↔ 𝑥 ∈ { 𝑥 } ) )
49 oveq2 ⊢ ( 𝑢 = { 𝑥 } → ( 𝒫 𝑋 ↾t 𝑢 ) = ( 𝒫 𝑋 ↾t { 𝑥 } ) )
50 49 eleq1d ⊢ ( 𝑢 = { 𝑥 } → ( ( 𝒫 𝑋 ↾t 𝑢 ) ∈ 𝐴 ↔ ( 𝒫 𝑋 ↾t { 𝑥 } ) ∈ 𝐴 ) )
51 48 50 anbi12d ⊢ ( 𝑢 = { 𝑥 } → ( ( 𝑥 ∈ 𝑢 ∧ ( 𝒫 𝑋 ↾t 𝑢 ) ∈ 𝐴 ) ↔ ( 𝑥 ∈ { 𝑥 } ∧ ( 𝒫 𝑋 ↾t { 𝑥 } ) ∈ 𝐴 ) ) )
52 51 rspcev ⊢ ( ( { 𝑥 } ∈ ( 𝒫 𝑋 ∩ 𝒫 𝑦 ) ∧ ( 𝑥 ∈ { 𝑥 } ∧ ( 𝒫 𝑋 ↾t { 𝑥 } ) ∈ 𝐴 ) ) → ∃ 𝑢 ∈ ( 𝒫 𝑋 ∩ 𝒫 𝑦 ) ( 𝑥 ∈ 𝑢 ∧ ( 𝒫 𝑋 ↾t 𝑢 ) ∈ 𝐴 ) )
53 41 43 47 52 syl12anc ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝒫 { 𝑥 } ∈ 𝐴 ) ) → ∃ 𝑢 ∈ ( 𝒫 𝑋 ∩ 𝒫 𝑦 ) ( 𝑥 ∈ 𝑢 ∧ ( 𝒫 𝑋 ↾t 𝑢 ) ∈ 𝐴 ) )
54 53 expr ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ 𝑥 ∈ 𝑦 ) → ( 𝒫 { 𝑥 } ∈ 𝐴 → ∃ 𝑢 ∈ ( 𝒫 𝑋 ∩ 𝒫 𝑦 ) ( 𝑥 ∈ 𝑢 ∧ ( 𝒫 𝑋 ↾t 𝑢 ) ∈ 𝐴 ) ) )
55 54 ralimdva ⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) → ( ∀ 𝑥 ∈ 𝑦 𝒫 { 𝑥 } ∈ 𝐴 → ∀ 𝑥 ∈ 𝑦 ∃ 𝑢 ∈ ( 𝒫 𝑋 ∩ 𝒫 𝑦 ) ( 𝑥 ∈ 𝑢 ∧ ( 𝒫 𝑋 ↾t 𝑢 ) ∈ 𝐴 ) ) )
56 31 55 syld ⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 𝒫 { 𝑥 } ∈ 𝐴 → ∀ 𝑥 ∈ 𝑦 ∃ 𝑢 ∈ ( 𝒫 𝑋 ∩ 𝒫 𝑦 ) ( 𝑥 ∈ 𝑢 ∧ ( 𝒫 𝑋 ↾t 𝑢 ) ∈ 𝐴 ) ) )
57 56 imp ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋 ) ∧ ∀ 𝑥 ∈ 𝑋 𝒫 { 𝑥 } ∈ 𝐴 ) → ∀ 𝑥 ∈ 𝑦 ∃ 𝑢 ∈ ( 𝒫 𝑋 ∩ 𝒫 𝑦 ) ( 𝑥 ∈ 𝑢 ∧ ( 𝒫 𝑋 ↾t 𝑢 ) ∈ 𝐴 ) )
58 57 an32s ⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 𝒫 { 𝑥 } ∈ 𝐴 ) ∧ 𝑦 ∈ 𝒫 𝑋 ) → ∀ 𝑥 ∈ 𝑦 ∃ 𝑢 ∈ ( 𝒫 𝑋 ∩ 𝒫 𝑦 ) ( 𝑥 ∈ 𝑢 ∧ ( 𝒫 𝑋 ↾t 𝑢 ) ∈ 𝐴 ) )
59 58 ralrimiva ⊢ ( ( 𝑋 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 𝒫 { 𝑥 } ∈ 𝐴 ) → ∀ 𝑦 ∈ 𝒫 𝑋 ∀ 𝑥 ∈ 𝑦 ∃ 𝑢 ∈ ( 𝒫 𝑋 ∩ 𝒫 𝑦 ) ( 𝑥 ∈ 𝑢 ∧ ( 𝒫 𝑋 ↾t 𝑢 ) ∈ 𝐴 ) )
60 islly ⊢ ( 𝒫 𝑋 ∈ Locally 𝐴 ↔ ( 𝒫 𝑋 ∈ Top ∧ ∀ 𝑦 ∈ 𝒫 𝑋 ∀ 𝑥 ∈ 𝑦 ∃ 𝑢 ∈ ( 𝒫 𝑋 ∩ 𝒫 𝑦 ) ( 𝑥 ∈ 𝑢 ∧ ( 𝒫 𝑋 ↾t 𝑢 ) ∈ 𝐴 ) ) )
61 27 59 60 sylanbrc ⊢ ( ( 𝑋 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝑋 𝒫 { 𝑥 } ∈ 𝐴 ) → 𝒫 𝑋 ∈ Locally 𝐴 )
62 25 61 impbida ⊢ ( 𝑋 ∈ 𝑉 → ( 𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀ 𝑥 ∈ 𝑋 𝒫 { 𝑥 } ∈ 𝐴 ) )