Theorem mremre 17656

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 17644	. . . . 5 ⊢ (𝑎 ∈ (Moore‘𝑋) → 𝑎 ⊆ 𝒫 𝑋)
2		velpw 4572	. . . . 5 ⊢ (𝑎 ∈ 𝒫 𝒫 𝑋 ↔ 𝑎 ⊆ 𝒫 𝑋)
3	1, 2	sylibr 237	. . . 4 ⊢ (𝑎 ∈ (Moore‘𝑋) → 𝑎 ∈ 𝒫 𝒫 𝑋)
4	3	ssriv 3949	. . 3 ⊢ (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋
5	4	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋)
6		ssidd 3968	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ⊆ 𝒫 𝑋)
7		pwidg 4587	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝒫 𝑋)
8		intssuni2 4942	. . . . . 6 ⊢ ((𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ ∪ 𝒫 𝑋)
9	8	3adant1 1146	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ ∪ 𝒫 𝑋)
10		unipw 5432	. . . . 5 ⊢ ∪ 𝒫 𝑋 = 𝑋
11	9, 10	sseqtrdi 3985	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ 𝑋)
12		elpw2g 5304	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∩ 𝑎 ∈ 𝒫 𝑋 ↔ ∩ 𝑎 ⊆ 𝑋))
13	12	3ad2ant1 1149	. . . 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 17654	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ (Moore‘𝑋))
16		n0 4315	. . . . 5 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝑎)
17		intss1 4932	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝑎 → ∩ 𝑎 ⊆ 𝑏)
18	17	adantl 486	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏 ∈ 𝑎) → ∩ 𝑎 ⊆ 𝑏)
19		simpr 489	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) → 𝑎 ⊆ (Moore‘𝑋))
20	19	sselda 3945	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ (Moore‘𝑋))
21		mresspw 17644	. . . . . . . . 9 ⊢ (𝑏 ∈ (Moore‘𝑋) → 𝑏 ⊆ 𝒫 𝑋)
22	20, 21	syl 18	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏 ∈ 𝑎) → 𝑏 ⊆ 𝒫 𝑋)
23	18, 22	sstrd 3955	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏 ∈ 𝑎) → ∩ 𝑎 ⊆ 𝒫 𝑋)
24	23	ex 417	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) → (𝑏 ∈ 𝑎 → ∩ 𝑎 ⊆ 𝒫 𝑋))
25	24	exlimdv 1960	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) → (∃𝑏 𝑏 ∈ 𝑎 → ∩ 𝑎 ⊆ 𝒫 𝑋))
26	16, 25	biimtrid 245	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) → (𝑎 ≠ ∅ → ∩ 𝑎 ⊆ 𝒫 𝑋))
27	26	3impia 1133	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ 𝒫 𝑋)
28		simp2 1153	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
29	28	sselda 3945	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ (Moore‘𝑋))
30		mre1cl 17646	. . . . . 6 ⊢ (𝑏 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝑏)
31	29, 30	syl 18	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ∈ 𝑎) → 𝑋 ∈ 𝑏)
32	31	ralrimiva 3163	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∀𝑏 ∈ 𝑎 𝑋 ∈ 𝑏)
33		elintg 4924	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ ∩ 𝑎 ↔ ∀𝑏 ∈ 𝑎 𝑋 ∈ 𝑏))
34	33	3ad2ant1 1149	. . . 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 1221	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
37	36	sselda 3945	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑐 ∈ (Moore‘𝑋))
38		simpl2 1209	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑏 ⊆ ∩ 𝑎)
39		intss1 4932	. . . . . . . 8 ⊢ (𝑐 ∈ 𝑎 → ∩ 𝑎 ⊆ 𝑐)
40	39	adantl 486	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → ∩ 𝑎 ⊆ 𝑐)
41	38, 40	sstrd 3955	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑏 ⊆ 𝑐)
42		simpl3 1210	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑏 ≠ ∅)
43		mreintcl 17647	. . . . . 6 ⊢ ((𝑐 ∈ (Moore‘𝑋) ∧ 𝑏 ⊆ 𝑐 ∧ 𝑏 ≠ ∅) → ∩ 𝑏 ∈ 𝑐)
44	37, 41, 42, 43	syl3anc 1396	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → ∩ 𝑏 ∈ 𝑐)
45	44	ralrimiva 3163	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) → ∀𝑐 ∈ 𝑎 ∩ 𝑏 ∈ 𝑐)
46		intex 5315	. . . . . 6 ⊢ (𝑏 ≠ ∅ ↔ ∩ 𝑏 ∈ V)
47		elintg 4924	. . . . . 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 1151	. . . 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 17654	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ∈ (Moore‘𝑋))
52	5, 15, 51	ismred 17654	1 ⊢ (𝑋 ∈ 𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  ∪ cuni 4876  ∩ cint 4916  ‘cfv 6537  Moorecmre 17634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-mre 17638

This theorem is referenced by:  mreacs  17714  mreclatdemoBAD  23222