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Theorem mremre 16863
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 16851 . . . . 5 (𝑎 ∈ (Moore‘𝑋) → 𝑎 ⊆ 𝒫 𝑋)
2 velpw 4543 . . . . 5 (𝑎 ∈ 𝒫 𝒫 𝑋𝑎 ⊆ 𝒫 𝑋)
31, 2sylibr 235 . . . 4 (𝑎 ∈ (Moore‘𝑋) → 𝑎 ∈ 𝒫 𝒫 𝑋)
43ssriv 3968 . . 3 (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋
54a1i 11 . 2 (𝑋𝑉 → (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋)
6 ssidd 3987 . . 3 (𝑋𝑉 → 𝒫 𝑋 ⊆ 𝒫 𝑋)
7 pwidg 4552 . . 3 (𝑋𝑉𝑋 ∈ 𝒫 𝑋)
8 intssuni2 4892 . . . . . 6 ((𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 𝒫 𝑋)
983adant1 1122 . . . . 5 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 𝒫 𝑋)
10 unipw 5333 . . . . 5 𝒫 𝑋 = 𝑋
119, 10sseqtrdi 4014 . . . 4 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎𝑋)
12 elpw2g 5238 . . . . 5 (𝑋𝑉 → ( 𝑎 ∈ 𝒫 𝑋 𝑎𝑋))
13123ad2ant1 1125 . . . 4 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → ( 𝑎 ∈ 𝒫 𝑋 𝑎𝑋))
1411, 13mpbird 258 . . 3 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 ∈ 𝒫 𝑋)
156, 7, 14ismred 16861 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ (Moore‘𝑋))
16 n0 4307 . . . . 5 (𝑎 ≠ ∅ ↔ ∃𝑏 𝑏𝑎)
17 intss1 4882 . . . . . . . . 9 (𝑏𝑎 𝑎𝑏)
1817adantl 482 . . . . . . . 8 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑎𝑏)
19 simpr 485 . . . . . . . . . 10 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → 𝑎 ⊆ (Moore‘𝑋))
2019sselda 3964 . . . . . . . . 9 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑏 ∈ (Moore‘𝑋))
21 mresspw 16851 . . . . . . . . 9 (𝑏 ∈ (Moore‘𝑋) → 𝑏 ⊆ 𝒫 𝑋)
2220, 21syl 17 . . . . . . . 8 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑏 ⊆ 𝒫 𝑋)
2318, 22sstrd 3974 . . . . . . 7 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑎 ⊆ 𝒫 𝑋)
2423ex 413 . . . . . 6 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (𝑏𝑎 𝑎 ⊆ 𝒫 𝑋))
2524exlimdv 1925 . . . . 5 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (∃𝑏 𝑏𝑎 𝑎 ⊆ 𝒫 𝑋))
2616, 25syl5bi 243 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (𝑎 ≠ ∅ → 𝑎 ⊆ 𝒫 𝑋))
27263impia 1109 . . 3 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ 𝒫 𝑋)
28 simp2 1129 . . . . . . 7 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
2928sselda 3964 . . . . . 6 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏𝑎) → 𝑏 ∈ (Moore‘𝑋))
30 mre1cl 16853 . . . . . 6 (𝑏 ∈ (Moore‘𝑋) → 𝑋𝑏)
3129, 30syl 17 . . . . 5 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏𝑎) → 𝑋𝑏)
3231ralrimiva 3179 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∀𝑏𝑎 𝑋𝑏)
33 elintg 4875 . . . . 5 (𝑋𝑉 → (𝑋 𝑎 ↔ ∀𝑏𝑎 𝑋𝑏))
34333ad2ant1 1125 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → (𝑋 𝑎 ↔ ∀𝑏𝑎 𝑋𝑏))
3532, 34mpbird 258 . . 3 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑋 𝑎)
36 simp12 1196 . . . . . . 7 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
3736sselda 3964 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑐 ∈ (Moore‘𝑋))
38 simpl2 1184 . . . . . . 7 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏 𝑎)
39 intss1 4882 . . . . . . . 8 (𝑐𝑎 𝑎𝑐)
4039adantl 482 . . . . . . 7 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑎𝑐)
4138, 40sstrd 3974 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏𝑐)
42 simpl3 1185 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏 ≠ ∅)
43 mreintcl 16854 . . . . . 6 ((𝑐 ∈ (Moore‘𝑋) ∧ 𝑏𝑐𝑏 ≠ ∅) → 𝑏𝑐)
4437, 41, 42, 43syl3anc 1363 . . . . 5 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏𝑐)
4544ralrimiva 3179 . . . 4 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → ∀𝑐𝑎 𝑏𝑐)
46 intex 5231 . . . . . 6 (𝑏 ≠ ∅ ↔ 𝑏 ∈ V)
47 elintg 4875 . . . . . 6 ( 𝑏 ∈ V → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
4846, 47sylbi 218 . . . . 5 (𝑏 ≠ ∅ → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
49483ad2ant3 1127 . . . 4 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
5045, 49mpbird 258 . . 3 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → 𝑏 𝑎)
5127, 35, 50ismred 16861 . 2 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ∈ (Moore‘𝑋))
525, 15, 51ismred 16861 1 (𝑋𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079  wex 1771  wcel 2105  wne 3013  wral 3135  Vcvv 3492  wss 3933  c0 4288  𝒫 cpw 4535   cuni 4830   cint 4867  cfv 6348  Moorecmre 16841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-mre 16845
This theorem is referenced by:  mreacs  16917  mreclatdemoBAD  21632
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