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Theorem mremre 16486
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 16474 . . . . 5 (𝑎 ∈ (Moore‘𝑋) → 𝑎 ⊆ 𝒫 𝑋)
2 selpw 4309 . . . . 5 (𝑎 ∈ 𝒫 𝒫 𝑋𝑎 ⊆ 𝒫 𝑋)
31, 2sylibr 224 . . . 4 (𝑎 ∈ (Moore‘𝑋) → 𝑎 ∈ 𝒫 𝒫 𝑋)
43ssriv 3748 . . 3 (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋
54a1i 11 . 2 (𝑋𝑉 → (Moore‘𝑋) ⊆ 𝒫 𝒫 𝑋)
6 ssid 3765 . . . 4 𝒫 𝑋 ⊆ 𝒫 𝑋
76a1i 11 . . 3 (𝑋𝑉 → 𝒫 𝑋 ⊆ 𝒫 𝑋)
8 pwidg 4317 . . 3 (𝑋𝑉𝑋 ∈ 𝒫 𝑋)
9 intssuni2 4654 . . . . . 6 ((𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 𝒫 𝑋)
1093adant1 1125 . . . . 5 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 𝒫 𝑋)
11 unipw 5067 . . . . 5 𝒫 𝑋 = 𝑋
1210, 11syl6sseq 3792 . . . 4 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎𝑋)
13 elpw2g 4976 . . . . 5 (𝑋𝑉 → ( 𝑎 ∈ 𝒫 𝑋 𝑎𝑋))
14133ad2ant1 1128 . . . 4 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → ( 𝑎 ∈ 𝒫 𝑋 𝑎𝑋))
1512, 14mpbird 247 . . 3 ((𝑋𝑉𝑎 ⊆ 𝒫 𝑋𝑎 ≠ ∅) → 𝑎 ∈ 𝒫 𝑋)
167, 8, 15ismred 16484 . 2 (𝑋𝑉 → 𝒫 𝑋 ∈ (Moore‘𝑋))
17 n0 4074 . . . . 5 (𝑎 ≠ ∅ ↔ ∃𝑏 𝑏𝑎)
18 intss1 4644 . . . . . . . . 9 (𝑏𝑎 𝑎𝑏)
1918adantl 473 . . . . . . . 8 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑎𝑏)
20 simpr 479 . . . . . . . . . 10 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → 𝑎 ⊆ (Moore‘𝑋))
2120sselda 3744 . . . . . . . . 9 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑏 ∈ (Moore‘𝑋))
22 mresspw 16474 . . . . . . . . 9 (𝑏 ∈ (Moore‘𝑋) → 𝑏 ⊆ 𝒫 𝑋)
2321, 22syl 17 . . . . . . . 8 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑏 ⊆ 𝒫 𝑋)
2419, 23sstrd 3754 . . . . . . 7 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) ∧ 𝑏𝑎) → 𝑎 ⊆ 𝒫 𝑋)
2524ex 449 . . . . . 6 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (𝑏𝑎 𝑎 ⊆ 𝒫 𝑋))
2625exlimdv 2010 . . . . 5 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (∃𝑏 𝑏𝑎 𝑎 ⊆ 𝒫 𝑋))
2717, 26syl5bi 232 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋)) → (𝑎 ≠ ∅ → 𝑎 ⊆ 𝒫 𝑋))
28273impia 1110 . . 3 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ 𝒫 𝑋)
29 simp2 1132 . . . . . . 7 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
3029sselda 3744 . . . . . 6 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏𝑎) → 𝑏 ∈ (Moore‘𝑋))
31 mre1cl 16476 . . . . . 6 (𝑏 ∈ (Moore‘𝑋) → 𝑋𝑏)
3230, 31syl 17 . . . . 5 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏𝑎) → 𝑋𝑏)
3332ralrimiva 3104 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → ∀𝑏𝑎 𝑋𝑏)
34 elintg 4635 . . . . 5 (𝑋𝑉 → (𝑋 𝑎 ↔ ∀𝑏𝑎 𝑋𝑏))
35343ad2ant1 1128 . . . 4 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → (𝑋 𝑎 ↔ ∀𝑏𝑎 𝑋𝑏))
3633, 35mpbird 247 . . 3 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑋 𝑎)
37 simp12 1247 . . . . . . 7 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → 𝑎 ⊆ (Moore‘𝑋))
3837sselda 3744 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑐 ∈ (Moore‘𝑋))
39 simpl2 1230 . . . . . . 7 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏 𝑎)
40 intss1 4644 . . . . . . . 8 (𝑐𝑎 𝑎𝑐)
4140adantl 473 . . . . . . 7 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑎𝑐)
4239, 41sstrd 3754 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏𝑐)
43 simpl3 1232 . . . . . 6 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏 ≠ ∅)
44 mreintcl 16477 . . . . . 6 ((𝑐 ∈ (Moore‘𝑋) ∧ 𝑏𝑐𝑏 ≠ ∅) → 𝑏𝑐)
4538, 42, 43, 44syl3anc 1477 . . . . 5 ((((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) ∧ 𝑐𝑎) → 𝑏𝑐)
4645ralrimiva 3104 . . . 4 (((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) ∧ 𝑏 𝑎𝑏 ≠ ∅) → ∀𝑐𝑎 𝑏𝑐)
47 intex 4969 . . . . . 6 (𝑏 ≠ ∅ ↔ 𝑏 ∈ V)
48 elintg 4635 . . . . . 6 ( 𝑏 ∈ V → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
4947, 48sylbi 207 . . . . 5 (𝑏 ≠ ∅ → ( 𝑏 𝑎 ↔ ∀𝑐𝑎 𝑏𝑐))
50493ad2ant3 1130 . . . 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 16484 . 2 ((𝑋𝑉𝑎 ⊆ (Moore‘𝑋) ∧ 𝑎 ≠ ∅) → 𝑎 ∈ (Moore‘𝑋))
535, 16, 52ismred 16484 1 (𝑋𝑉 → (Moore‘𝑋) ∈ (Moore‘𝒫 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wex 1853  wcel 2139  wne 2932  wral 3050  Vcvv 3340  wss 3715  c0 4058  𝒫 cpw 4302   cuni 4588   cint 4627  cfv 6049  Moorecmre 16464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-mre 16468
This theorem is referenced by:  mreacs  16540  mreclatdemoBAD  21122
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