Theorem mremre 17107

Description: The Moore collections of subsets of a space, viewed as a kind of subset of the power set, form a Moore collection in their own right on the power set. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Assertion

Ref	Expression
mremre	⊢ (𝑋 ∈ 𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))

Proof of Theorem mremre

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mresspw 17095	. . . . 5 ⊢ (𝑎 ∈ (Moore‘𝑋) → 𝑎 ⊆ 𝒫 𝑋)
2		velpw 4518	. . . . 5 ⊢ (𝑎 ∈ 𝒫 𝒫 𝑋 ↔ 𝑎 ⊆ 𝒫 𝑋)
3	1, 2	sylibr 237	. . . 4 ⊢ (𝑎 ∈ (Moore‘𝑋) → 𝑎 ∈ 𝒫 𝒫 𝑋)
4	3	ssriv 3905	. . 3 ⊢ (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋
5	4	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋)
6		ssidd 3924	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ⊆ 𝒫 𝑋)
7		pwidg 4535	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝒫 𝑋)
8		intssuni2 4884	. . . . . 6 ⊢ ((𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ ∪ 𝒫 𝑋)
9	8	3adant1 1132	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ ∪ 𝒫 𝑋)
10		unipw 5335	. . . . 5 ⊢ ∪ 𝒫 𝑋 = 𝑋
11	9, 10	sseqtrdi 3951	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ 𝑋)
12		elpw2g 5237	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∩ 𝑎 ∈ 𝒫 𝑋 ↔ ∩ 𝑎 ⊆ 𝑋))
13	12	3ad2ant1 1135	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → (∩ 𝑎 ∈ 𝒫 𝑋 ↔ ∩ 𝑎 ⊆ 𝑋))
14	11, 13	mpbird 260	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ∈ 𝒫 𝑋)
15	6, 7, 14	ismred 17105	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ (Moore‘𝑋))
16		n0 4261	. . . . 5 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝑎)
17		intss1 4874	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝑎 → ∩ 𝑎 ⊆ 𝑏)
18	17	adantl 485	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏 ∈ 𝑎) → ∩ 𝑎 ⊆ 𝑏)
19		simpr 488	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) → 𝑎 ⊆ (Moore‘𝑋))
20	19	sselda 3901	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ (Moore‘𝑋))
21		mresspw 17095	. . . . . . . . 9 ⊢ (𝑏 ∈ (Moore‘𝑋) → 𝑏 ⊆ 𝒫 𝑋)
22	20, 21	syl 17	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏 ∈ 𝑎) → 𝑏 ⊆ 𝒫 𝑋)
23	18, 22	sstrd 3911	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏 ∈ 𝑎) → ∩ 𝑎 ⊆ 𝒫 𝑋)
24	23	ex 416	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) → (𝑏 ∈ 𝑎 → ∩ 𝑎 ⊆ 𝒫 𝑋))
25	24	exlimdv 1941	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) → (∃𝑏 𝑏 ∈ 𝑎 → ∩ 𝑎 ⊆ 𝒫 𝑋))
26	16, 25	syl5bi 245	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) → (𝑎 ≠ ∅ → ∩ 𝑎 ⊆ 𝒫 𝑋))
27	26	3impia 1119	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ 𝒫 𝑋)
28		simp2 1139	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
29	28	sselda 3901	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ (Moore‘𝑋))
30		mre1cl 17097	. . . . . 6 ⊢ (𝑏 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝑏)
31	29, 30	syl 17	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ∈ 𝑎) → 𝑋 ∈ 𝑏)
32	31	ralrimiva 3105	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∀𝑏 ∈ 𝑎 𝑋 ∈ 𝑏)
33		elintg 4867	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ ∩ 𝑎 ↔ ∀𝑏 ∈ 𝑎 𝑋 ∈ 𝑏))
34	33	3ad2ant1 1135	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → (𝑋 ∈ ∩ 𝑎 ↔ ∀𝑏 ∈ 𝑎 𝑋 ∈ 𝑏))
35	32, 34	mpbird 260	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑋 ∈ ∩ 𝑎)
36		simp12 1206	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
37	36	sselda 3901	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑐 ∈ (Moore‘𝑋))
38		simpl2 1194	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑏 ⊆ ∩ 𝑎)
39		intss1 4874	. . . . . . . 8 ⊢ (𝑐 ∈ 𝑎 → ∩ 𝑎 ⊆ 𝑐)
40	39	adantl 485	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → ∩ 𝑎 ⊆ 𝑐)
41	38, 40	sstrd 3911	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑏 ⊆ 𝑐)
42		simpl3 1195	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑏 ≠ ∅)
43		mreintcl 17098	. . . . . 6 ⊢ ((𝑐 ∈ (Moore‘𝑋) ∧ 𝑏 ⊆ 𝑐 ∧ 𝑏 ≠ ∅) → ∩ 𝑏 ∈ 𝑐)
44	37, 41, 42, 43	syl3anc 1373	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → ∩ 𝑏 ∈ 𝑐)
45	44	ralrimiva 3105	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) → ∀𝑐 ∈ 𝑎 ∩ 𝑏 ∈ 𝑐)
46		intex 5230	. . . . . 6 ⊢ (𝑏 ≠ ∅ ↔ ∩ 𝑏 ∈ V)
47		elintg 4867	. . . . . 6 ⊢ (∩ 𝑏 ∈ V → (∩ 𝑏 ∈ ∩ 𝑎 ↔ ∀𝑐 ∈ 𝑎 ∩ 𝑏 ∈ 𝑐))
48	46, 47	sylbi 220	. . . . 5 ⊢ (𝑏 ≠ ∅ → (∩ 𝑏 ∈ ∩ 𝑎 ↔ ∀𝑐 ∈ 𝑎 ∩ 𝑏 ∈ 𝑐))
49	48	3ad2ant3 1137	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) → (∩ 𝑏 ∈ ∩ 𝑎 ↔ ∀𝑐 ∈ 𝑎 ∩ 𝑏 ∈ 𝑐))
50	45, 49	mpbird 260	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) → ∩ 𝑏 ∈ ∩ 𝑎)
51	27, 35, 50	ismred 17105	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ∈ (Moore‘𝑋))
52	5, 15, 51	ismred 17105	1 ⊢ (𝑋 ∈ 𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089  ∃wex 1787   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  Vcvv 3408   ⊆ wss 3866  ∅c0 4237  𝒫 cpw 4513  ∪ cuni 4819  ∩ cint 4859  ‘cfv 6380  Moorecmre 17085

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-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322

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-rab 3070  df-v 3410  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-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-iota 6338  df-fun 6382  df-fv 6388  df-mre 17089

This theorem is referenced by:  mreacs  17161  mreclatdemoBAD  21993