Theorem mremre 17651

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 17639	. . . . 5 ⊢ (𝑎 ∈ (Moore‘𝑋) → 𝑎 ⊆ 𝒫 𝑋)
2		velpw 4567	. . . . 5 ⊢ (𝑎 ∈ 𝒫 𝒫 𝑋 ↔ 𝑎 ⊆ 𝒫 𝑋)
3	1, 2	sylibr 237	. . . 4 ⊢ (𝑎 ∈ (Moore‘𝑋) → 𝑎 ∈ 𝒫 𝒫 𝑋)
4	3	ssriv 3941	. . 3 ⊢ (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋
5	4	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋)
6		ssidd 3960	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ⊆ 𝒫 𝑋)
7		pwidg 4582	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝒫 𝑋)
8		intssuni2 4938	. . . . . 6 ⊢ ((𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ ∪ 𝒫 𝑋)
9	8	3adant1 1148	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ ∪ 𝒫 𝑋)
10		unipw 5431	. . . . 5 ⊢ ∪ 𝒫 𝑋 = 𝑋
11	9, 10	sseqtrdi 3977	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ 𝑋)
12		elpw2g 5304	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∩ 𝑎 ∈ 𝒫 𝑋 ↔ ∩ 𝑎 ⊆ 𝑋))
13	12	3ad2ant1 1151	. . . 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 17649	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ (Moore‘𝑋))
16		n0 4307	. . . . 5 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝑎)
17		intss1 4928	. . . . . . . . 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 3937	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ (Moore‘𝑋))
21		mresspw 17639	. . . . . . . . 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 3947	. . . . . . 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 1963	. . . . 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 1135	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ 𝒫 𝑋)
28		simp2 1155	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
29	28	sselda 3937	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ (Moore‘𝑋))
30		mre1cl 17641	. . . . . 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 3157	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∀𝑏 ∈ 𝑎 𝑋 ∈ 𝑏)
33		elintg 4920	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ ∩ 𝑎 ↔ ∀𝑏 ∈ 𝑎 𝑋 ∈ 𝑏))
34	33	3ad2ant1 1151	. . . 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 1223	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
37	36	sselda 3937	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑐 ∈ (Moore‘𝑋))
38		simpl2 1211	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑏 ⊆ ∩ 𝑎)
39		intss1 4928	. . . . . . . 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 3947	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑏 ⊆ 𝑐)
42		simpl3 1212	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑏 ≠ ∅)
43		mreintcl 17642	. . . . . 6 ⊢ ((𝑐 ∈ (Moore‘𝑋) ∧ 𝑏 ⊆ 𝑐 ∧ 𝑏 ≠ ∅) → ∩ 𝑏 ∈ 𝑐)
44	37, 41, 42, 43	syl3anc 1398	. . . . 5 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → ∩ 𝑏 ∈ 𝑐)
45	44	ralrimiva 3157	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) → ∀𝑐 ∈ 𝑎 ∩ 𝑏 ∈ 𝑐)
46		intex 5314	. . . . . 6 ⊢ (𝑏 ≠ ∅ ↔ ∩ 𝑏 ∈ V)
47		elintg 4920	. . . . . 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 1153	. . . 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 17649	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ∈ (Moore‘𝑋))
52	5, 15, 51	ismred 17649	1 ⊢ (𝑋 ∈ 𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4562  ∪ cuni 4872  ∩ cint 4912  ‘cfv 6536  Moorecmre 17629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-mre 17633

This theorem is referenced by:  mreacs  17709  mreclatdemoBAD  23253