Theorem mremre 17523

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 17511	. . . . 5 ⊢ (𝑎 ∈ (Moore‘𝑋) → 𝑎 ⊆ 𝒫 𝑋)
2		velpw 4559	. . . . 5 ⊢ (𝑎 ∈ 𝒫 𝒫 𝑋 ↔ 𝑎 ⊆ 𝒫 𝑋)
3	1, 2	sylibr 234	. . . 4 ⊢ (𝑎 ∈ (Moore‘𝑋) → 𝑎 ∈ 𝒫 𝒫 𝑋)
4	3	ssriv 3937	. . 3 ⊢ (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋
5	4	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋)
6		ssidd 3957	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ⊆ 𝒫 𝑋)
7		pwidg 4574	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝒫 𝑋)
8		intssuni2 4928	. . . . . 6 ⊢ ((𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ ∪ 𝒫 𝑋)
9	8	3adant1 1130	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ ∪ 𝒫 𝑋)
10		unipw 5398	. . . . 5 ⊢ ∪ 𝒫 𝑋 = 𝑋
11	9, 10	sseqtrdi 3974	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ 𝑋)
12		elpw2g 5278	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∩ 𝑎 ∈ 𝒫 𝑋 ↔ ∩ 𝑎 ⊆ 𝑋))
13	12	3ad2ant1 1133	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → (∩ 𝑎 ∈ 𝒫 𝑋 ↔ ∩ 𝑎 ⊆ 𝑋))
14	11, 13	mpbird 257	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ 𝒫 𝑋 ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ∈ 𝒫 𝑋)
15	6, 7, 14	ismred 17521	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ (Moore‘𝑋))
16		n0 4305	. . . . 5 ⊢ (𝑎 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝑎)
17		intss1 4918	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝑎 → ∩ 𝑎 ⊆ 𝑏)
18	17	adantl 481	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏 ∈ 𝑎) → ∩ 𝑎 ⊆ 𝑏)
19		simpr 484	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) → 𝑎 ⊆ (Moore‘𝑋))
20	19	sselda 3933	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ (Moore‘𝑋))
21		mresspw 17511	. . . . . . . . 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 3944	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏 ∈ 𝑎) → ∩ 𝑎 ⊆ 𝒫 𝑋)
24	23	ex 412	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) → (𝑏 ∈ 𝑎 → ∩ 𝑎 ⊆ 𝒫 𝑋))
25	24	exlimdv 1934	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) → (∃𝑏 𝑏 ∈ 𝑎 → ∩ 𝑎 ⊆ 𝒫 𝑋))
26	16, 25	biimtrid 242	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋)) → (𝑎 ≠ ∅ → ∩ 𝑎 ⊆ 𝒫 𝑋))
27	26	3impia 1117	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ⊆ 𝒫 𝑋)
28		simp2 1137	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
29	28	sselda 3933	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ (Moore‘𝑋))
30		mre1cl 17513	. . . . . 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 3128	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∀𝑏 ∈ 𝑎 𝑋 ∈ 𝑏)
33		elintg 4910	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ ∩ 𝑎 ↔ ∀𝑏 ∈ 𝑎 𝑋 ∈ 𝑏))
34	33	3ad2ant1 1133	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → (𝑋 ∈ ∩ 𝑎 ↔ ∀𝑏 ∈ 𝑎 𝑋 ∈ 𝑏))
35	32, 34	mpbird 257	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑋 ∈ ∩ 𝑎)
36		simp12 1205	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
37	36	sselda 3933	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑐 ∈ (Moore‘𝑋))
38		simpl2 1193	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑏 ⊆ ∩ 𝑎)
39		intss1 4918	. . . . . . . 8 ⊢ (𝑐 ∈ 𝑎 → ∩ 𝑎 ⊆ 𝑐)
40	39	adantl 481	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → ∩ 𝑎 ⊆ 𝑐)
41	38, 40	sstrd 3944	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑏 ⊆ 𝑐)
42		simpl3 1194	. . . . . 6 ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) ∧ 𝑐 ∈ 𝑎) → 𝑏 ≠ ∅)
43		mreintcl 17514	. . . . . 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 3128	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) → ∀𝑐 ∈ 𝑎 ∩ 𝑏 ∈ 𝑐)
46		intex 5289	. . . . . 6 ⊢ (𝑏 ≠ ∅ ↔ ∩ 𝑏 ∈ V)
47		elintg 4910	. . . . . 6 ⊢ (∩ 𝑏 ∈ V → (∩ 𝑏 ∈ ∩ 𝑎 ↔ ∀𝑐 ∈ 𝑎 ∩ 𝑏 ∈ 𝑐))
48	46, 47	sylbi 217	. . . . 5 ⊢ (𝑏 ≠ ∅ → (∩ 𝑏 ∈ ∩ 𝑎 ↔ ∀𝑐 ∈ 𝑎 ∩ 𝑏 ∈ 𝑐))
49	48	3ad2ant3 1135	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) → (∩ 𝑏 ∈ ∩ 𝑎 ↔ ∀𝑐 ∈ 𝑎 ∩ 𝑏 ∈ 𝑐))
50	45, 49	mpbird 257	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 ⊆ ∩ 𝑎 ∧ 𝑏 ≠ ∅) → ∩ 𝑏 ∈ ∩ 𝑎)
51	27, 35, 50	ismred 17521	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∩ 𝑎 ∈ (Moore‘𝑋))
52	5, 15, 51	ismred 17521	1 ⊢ (𝑋 ∈ 𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  ∪ cuni 4863  ∩ cint 4902  ‘cfv 6492  Moorecmre 17501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-mre 17505

This theorem is referenced by:  mreacs  17581  mreclatdemoBAD  23040