MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  submre Structured version   Visualization version   GIF version

Theorem submre 16864
Description: The subcollection of a closed set system below a given closed set is itself a closed set system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Assertion
Ref Expression
submre ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))

Proof of Theorem submre
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 inss2 4203 . . 3 (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
21a1i 11 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
3 simpr 485 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴𝐶)
4 pwidg 4552 . . . 4 (𝐴𝐶𝐴 ∈ 𝒫 𝐴)
54adantl 482 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴 ∈ 𝒫 𝐴)
63, 5elind 4168 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴 ∈ (𝐶 ∩ 𝒫 𝐴))
7 simp1l 1189 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
8 inss1 4202 . . . . . 6 (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶
9 sstr 3972 . . . . . 6 ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶) → 𝑥𝐶)
108, 9mpan2 687 . . . . 5 (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥𝐶)
11103ad2ant2 1126 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐶)
12 simp3 1130 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ≠ ∅)
13 mreintcl 16854 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥𝐶𝑥 ≠ ∅) → 𝑥𝐶)
147, 11, 12, 13syl3anc 1363 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐶)
15 sstr 3972 . . . . . . . 8 ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
161, 15mpan2 687 . . . . . . 7 (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
17163ad2ant2 1126 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝒫 𝐴)
18 intssuni2 4892 . . . . . 6 ((𝑥 ⊆ 𝒫 𝐴𝑥 ≠ ∅) → 𝑥 𝒫 𝐴)
1917, 12, 18syl2anc 584 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 𝒫 𝐴)
20 unipw 5333 . . . . 5 𝒫 𝐴 = 𝐴
2119, 20sseqtrdi 4014 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐴)
22 elpw2g 5238 . . . . . 6 (𝐴𝐶 → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
2322adantl 482 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
24233ad2ant1 1125 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
2521, 24mpbird 258 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ 𝒫 𝐴)
2614, 25elind 4168 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (𝐶 ∩ 𝒫 𝐴))
272, 6, 26ismred 16861 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079  wcel 2105  wne 3013  cin 3932  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:  submrc  16887
  Copyright terms: Public domain W3C validator