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Theorem mremre 16185
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 16173 . . . . 5 (𝑎 ∈ (Moore‘𝑋) → 𝑎 ⊆ 𝒫 𝑋)
2 selpw 4137 . . . . 5 (𝑎 ∈ 𝒫 𝒫 𝑋𝑎 ⊆ 𝒫 𝑋)
31, 2sylibr 224 . . . 4 (𝑎 ∈ (Moore‘𝑋) → 𝑎 ∈ 𝒫 𝒫 𝑋)
43ssriv 3587 . . 3 (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋
54a1i 11 . 2 (𝑋𝑉 → (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋)
6 ssid 3603 . . . 4 𝒫 𝑋 ⊆ 𝒫 𝑋
76a1i 11 . . 3 (𝑋𝑉 → 𝒫 𝑋 ⊆ 𝒫 𝑋)
8 pwidg 4144 . . 3 (𝑋𝑉𝑋 ∈ 𝒫 𝑋)
9 intssuni2 4467 . . . . . 6 ((𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 𝒫 𝑋)
1093adant1 1077 . . . . 5 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 𝒫 𝑋)
11 unipw 4879 . . . . 5 𝒫 𝑋 = 𝑋
1210, 11syl6sseq 3630 . . . 4 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎𝑋)
13 elpw2g 4787 . . . . 5 (𝑋𝑉 → ( 𝑎 ∈ 𝒫 𝑋 𝑎𝑋))
14133ad2ant1 1080 . . . 4 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → ( 𝑎 ∈ 𝒫 𝑋 𝑎𝑋))
1512, 14mpbird 247 . . 3 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 ∈ 𝒫 𝑋)
167, 8, 15ismred 16183 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ (Moore‘𝑋))
17 n0 3907 . . . . 5 (𝑎 ≠ ∅ ↔ ∃𝑏 𝑏𝑎)
18 intss1 4457 . . . . . . . . 9 (𝑏𝑎 𝑎𝑏)
1918adantl 482 . . . . . . . 8 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑎𝑏)
20 simpr 477 . . . . . . . . . 10 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → 𝑎 ⊆ (Moore‘𝑋))
2120sselda 3583 . . . . . . . . 9 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑏 ∈ (Moore‘𝑋))
22 mresspw 16173 . . . . . . . . 9 (𝑏 ∈ (Moore‘𝑋) → 𝑏 ⊆ 𝒫 𝑋)
2321, 22syl 17 . . . . . . . 8 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑏 ⊆ 𝒫 𝑋)
2419, 23sstrd 3593 . . . . . . 7 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑎 ⊆ 𝒫 𝑋)
2524ex 450 . . . . . 6 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (𝑏𝑎 𝑎 ⊆ 𝒫 𝑋))
2625exlimdv 1858 . . . . 5 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (∃𝑏 𝑏𝑎 𝑎 ⊆ 𝒫 𝑋))
2717, 26syl5bi 232 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (𝑎 ≠ ∅ → 𝑎 ⊆ 𝒫 𝑋))
28273impia 1258 . . 3 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ 𝒫 𝑋)
29 simp2 1060 . . . . . . 7 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
3029sselda 3583 . . . . . 6 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏𝑎) → 𝑏 ∈ (Moore‘𝑋))
31 mre1cl 16175 . . . . . 6 (𝑏 ∈ (Moore‘𝑋) → 𝑋𝑏)
3230, 31syl 17 . . . . 5 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏𝑎) → 𝑋𝑏)
3332ralrimiva 2960 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∀𝑏𝑎 𝑋𝑏)
34 elintg 4448 . . . . 5 (𝑋𝑉 → (𝑋 𝑎 ↔ ∀𝑏𝑎 𝑋𝑏))
35343ad2ant1 1080 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → (𝑋 𝑎 ↔ ∀𝑏𝑎 𝑋𝑏))
3633, 35mpbird 247 . . 3 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑋 𝑎)
37 simp12 1090 . . . . . . 7 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
3837sselda 3583 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑐 ∈ (Moore‘𝑋))
39 simpl2 1063 . . . . . . 7 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏 𝑎)
40 intss1 4457 . . . . . . . 8 (𝑐𝑎 𝑎𝑐)
4140adantl 482 . . . . . . 7 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑎𝑐)
4239, 41sstrd 3593 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏𝑐)
43 simpl3 1064 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏 ≠ ∅)
44 mreintcl 16176 . . . . . 6 ((𝑐 ∈ (Moore‘𝑋) ∧ 𝑏𝑐𝑏 ≠ ∅) → 𝑏𝑐)
4538, 42, 43, 44syl3anc 1323 . . . . 5 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏𝑐)
4645ralrimiva 2960 . . . 4 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → ∀𝑐𝑎 𝑏𝑐)
47 intex 4780 . . . . . 6 (𝑏 ≠ ∅ ↔ 𝑏 ∈ V)
48 elintg 4448 . . . . . 6 ( 𝑏 ∈ V → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
4947, 48sylbi 207 . . . . 5 (𝑏 ≠ ∅ → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
50493ad2ant3 1082 . . . 4 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
5146, 50mpbird 247 . . 3 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → 𝑏 𝑎)
5228, 36, 51ismred 16183 . 2 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ∈ (Moore‘𝑋))
535, 16, 52ismred 16183 1 (𝑋𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wex 1701  wcel 1987  wne 2790  wral 2907  Vcvv 3186  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402   cint 4440  cfv 5847  Moorecmre 16163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-mre 16167
This theorem is referenced by:  mreacs  16240  mreclatdemoBAD  20810
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