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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 (𝑋𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))

Proof of Theorem mremre
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mresspw 17095 . . . . 5 (𝑎 ∈ (Moore‘𝑋) → 𝑎 ⊆ 𝒫 𝑋)
2 velpw 4518 . . . . 5 (𝑎 ∈ 𝒫 𝒫 𝑋𝑎 ⊆ 𝒫 𝑋)
31, 2sylibr 237 . . . 4 (𝑎 ∈ (Moore‘𝑋) → 𝑎 ∈ 𝒫 𝒫 𝑋)
43ssriv 3905 . . 3 (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋
54a1i 11 . 2 (𝑋𝑉 → (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋)
6 ssidd 3924 . . 3 (𝑋𝑉 → 𝒫 𝑋 ⊆ 𝒫 𝑋)
7 pwidg 4535 . . 3 (𝑋𝑉𝑋 ∈ 𝒫 𝑋)
8 intssuni2 4884 . . . . . 6 ((𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 𝒫 𝑋)
983adant1 1132 . . . . 5 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 𝒫 𝑋)
10 unipw 5335 . . . . 5 𝒫 𝑋 = 𝑋
119, 10sseqtrdi 3951 . . . 4 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎𝑋)
12 elpw2g 5237 . . . . 5 (𝑋𝑉 → ( 𝑎 ∈ 𝒫 𝑋 𝑎𝑋))
13123ad2ant1 1135 . . . 4 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → ( 𝑎 ∈ 𝒫 𝑋 𝑎𝑋))
1411, 13mpbird 260 . . 3 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 ∈ 𝒫 𝑋)
156, 7, 14ismred 17105 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ (Moore‘𝑋))
16 n0 4261 . . . . 5 (𝑎 ≠ ∅ ↔ ∃𝑏 𝑏𝑎)
17 intss1 4874 . . . . . . . . 9 (𝑏𝑎 𝑎𝑏)
1817adantl 485 . . . . . . . 8 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑎𝑏)
19 simpr 488 . . . . . . . . . 10 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → 𝑎 ⊆ (Moore‘𝑋))
2019sselda 3901 . . . . . . . . 9 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑏 ∈ (Moore‘𝑋))
21 mresspw 17095 . . . . . . . . 9 (𝑏 ∈ (Moore‘𝑋) → 𝑏 ⊆ 𝒫 𝑋)
2220, 21syl 17 . . . . . . . 8 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑏 ⊆ 𝒫 𝑋)
2318, 22sstrd 3911 . . . . . . 7 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑎 ⊆ 𝒫 𝑋)
2423ex 416 . . . . . 6 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (𝑏𝑎 𝑎 ⊆ 𝒫 𝑋))
2524exlimdv 1941 . . . . 5 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (∃𝑏 𝑏𝑎 𝑎 ⊆ 𝒫 𝑋))
2616, 25syl5bi 245 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (𝑎 ≠ ∅ → 𝑎 ⊆ 𝒫 𝑋))
27263impia 1119 . . 3 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ 𝒫 𝑋)
28 simp2 1139 . . . . . . 7 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
2928sselda 3901 . . . . . 6 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏𝑎) → 𝑏 ∈ (Moore‘𝑋))
30 mre1cl 17097 . . . . . 6 (𝑏 ∈ (Moore‘𝑋) → 𝑋𝑏)
3129, 30syl 17 . . . . 5 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏𝑎) → 𝑋𝑏)
3231ralrimiva 3105 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∀𝑏𝑎 𝑋𝑏)
33 elintg 4867 . . . . 5 (𝑋𝑉 → (𝑋 𝑎 ↔ ∀𝑏𝑎 𝑋𝑏))
34333ad2ant1 1135 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → (𝑋 𝑎 ↔ ∀𝑏𝑎 𝑋𝑏))
3532, 34mpbird 260 . . 3 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑋 𝑎)
36 simp12 1206 . . . . . . 7 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
3736sselda 3901 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑐 ∈ (Moore‘𝑋))
38 simpl2 1194 . . . . . . 7 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏 𝑎)
39 intss1 4874 . . . . . . . 8 (𝑐𝑎 𝑎𝑐)
4039adantl 485 . . . . . . 7 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑎𝑐)
4138, 40sstrd 3911 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏𝑐)
42 simpl3 1195 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏 ≠ ∅)
43 mreintcl 17098 . . . . . 6 ((𝑐 ∈ (Moore‘𝑋) ∧ 𝑏𝑐𝑏 ≠ ∅) → 𝑏𝑐)
4437, 41, 42, 43syl3anc 1373 . . . . 5 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏𝑐)
4544ralrimiva 3105 . . . 4 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → ∀𝑐𝑎 𝑏𝑐)
46 intex 5230 . . . . . 6 (𝑏 ≠ ∅ ↔ 𝑏 ∈ V)
47 elintg 4867 . . . . . 6 ( 𝑏 ∈ V → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
4846, 47sylbi 220 . . . . 5 (𝑏 ≠ ∅ → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
49483ad2ant3 1137 . . . 4 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
5045, 49mpbird 260 . . 3 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → 𝑏 𝑎)
5127, 35, 50ismred 17105 . 2 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ∈ (Moore‘𝑋))
525, 15, 51ismred 17105 1 (𝑋𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))
Colors of variables: wff setvar class
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
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