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Theorem mremre 16544
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 16532 . . . . 5 (𝑎 ∈ (Moore‘𝑋) → 𝑎 ⊆ 𝒫 𝑋)
2 selpw 4324 . . . . 5 (𝑎 ∈ 𝒫 𝒫 𝑋𝑎 ⊆ 𝒫 𝑋)
31, 2sylibr 225 . . . 4 (𝑎 ∈ (Moore‘𝑋) → 𝑎 ∈ 𝒫 𝒫 𝑋)
43ssriv 3767 . . 3 (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋
54a1i 11 . 2 (𝑋𝑉 → (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋)
6 ssidd 3786 . . 3 (𝑋𝑉 → 𝒫 𝑋 ⊆ 𝒫 𝑋)
7 pwidg 4332 . . 3 (𝑋𝑉𝑋 ∈ 𝒫 𝑋)
8 intssuni2 4660 . . . . . 6 ((𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 𝒫 𝑋)
983adant1 1160 . . . . 5 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 𝒫 𝑋)
10 unipw 5076 . . . . 5 𝒫 𝑋 = 𝑋
119, 10syl6sseq 3813 . . . 4 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎𝑋)
12 elpw2g 4987 . . . . 5 (𝑋𝑉 → ( 𝑎 ∈ 𝒫 𝑋 𝑎𝑋))
13123ad2ant1 1163 . . . 4 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → ( 𝑎 ∈ 𝒫 𝑋 𝑎𝑋))
1411, 13mpbird 248 . . 3 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 ∈ 𝒫 𝑋)
156, 7, 14ismred 16542 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ (Moore‘𝑋))
16 n0 4097 . . . . 5 (𝑎 ≠ ∅ ↔ ∃𝑏 𝑏𝑎)
17 intss1 4650 . . . . . . . . 9 (𝑏𝑎 𝑎𝑏)
1817adantl 473 . . . . . . . 8 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑎𝑏)
19 simpr 477 . . . . . . . . . 10 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → 𝑎 ⊆ (Moore‘𝑋))
2019sselda 3763 . . . . . . . . 9 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑏 ∈ (Moore‘𝑋))
21 mresspw 16532 . . . . . . . . 9 (𝑏 ∈ (Moore‘𝑋) → 𝑏 ⊆ 𝒫 𝑋)
2220, 21syl 17 . . . . . . . 8 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑏 ⊆ 𝒫 𝑋)
2318, 22sstrd 3773 . . . . . . 7 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑎 ⊆ 𝒫 𝑋)
2423ex 401 . . . . . 6 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (𝑏𝑎 𝑎 ⊆ 𝒫 𝑋))
2524exlimdv 2028 . . . . 5 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (∃𝑏 𝑏𝑎 𝑎 ⊆ 𝒫 𝑋))
2616, 25syl5bi 233 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (𝑎 ≠ ∅ → 𝑎 ⊆ 𝒫 𝑋))
27263impia 1145 . . 3 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ 𝒫 𝑋)
28 simp2 1167 . . . . . . 7 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
2928sselda 3763 . . . . . 6 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏𝑎) → 𝑏 ∈ (Moore‘𝑋))
30 mre1cl 16534 . . . . . 6 (𝑏 ∈ (Moore‘𝑋) → 𝑋𝑏)
3129, 30syl 17 . . . . 5 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏𝑎) → 𝑋𝑏)
3231ralrimiva 3113 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∀𝑏𝑎 𝑋𝑏)
33 elintg 4643 . . . . 5 (𝑋𝑉 → (𝑋 𝑎 ↔ ∀𝑏𝑎 𝑋𝑏))
34333ad2ant1 1163 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → (𝑋 𝑎 ↔ ∀𝑏𝑎 𝑋𝑏))
3532, 34mpbird 248 . . 3 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑋 𝑎)
36 simp12 1261 . . . . . . 7 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
3736sselda 3763 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑐 ∈ (Moore‘𝑋))
38 simpl2 1244 . . . . . . 7 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏 𝑎)
39 intss1 4650 . . . . . . . 8 (𝑐𝑎 𝑎𝑐)
4039adantl 473 . . . . . . 7 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑎𝑐)
4138, 40sstrd 3773 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏𝑐)
42 simpl3 1246 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏 ≠ ∅)
43 mreintcl 16535 . . . . . 6 ((𝑐 ∈ (Moore‘𝑋) ∧ 𝑏𝑐𝑏 ≠ ∅) → 𝑏𝑐)
4437, 41, 42, 43syl3anc 1490 . . . . 5 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏𝑐)
4544ralrimiva 3113 . . . 4 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → ∀𝑐𝑎 𝑏𝑐)
46 intex 4980 . . . . . 6 (𝑏 ≠ ∅ ↔ 𝑏 ∈ V)
47 elintg 4643 . . . . . 6 ( 𝑏 ∈ V → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
4846, 47sylbi 208 . . . . 5 (𝑏 ≠ ∅ → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
49483ad2ant3 1165 . . . 4 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
5045, 49mpbird 248 . . 3 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → 𝑏 𝑎)
5127, 35, 50ismred 16542 . 2 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ∈ (Moore‘𝑋))
525, 15, 51ismred 16542 1 (𝑋𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107  wex 1874  wcel 2155  wne 2937  wral 3055  Vcvv 3350  wss 3734  c0 4081  𝒫 cpw 4317   cuni 4596   cint 4635  cfv 6070  Moorecmre 16522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-int 4636  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-iota 6033  df-fun 6072  df-fv 6078  df-mre 16526
This theorem is referenced by:  mreacs  16598  mreclatdemoBAD  21194
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