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Theorem submre 16088
 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 3796 . . 3 (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
21a1i 11 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
3 simpr 476 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴𝐶)
4 pwidg 4121 . . . 4 (𝐴𝐶𝐴 ∈ 𝒫 𝐴)
54adantl 481 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴 ∈ 𝒫 𝐴)
63, 5elind 3760 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴 ∈ (𝐶 ∩ 𝒫 𝐴))
7 simp1l 1078 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
8 inss1 3795 . . . . . 6 (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶
9 sstr 3576 . . . . . 6 ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶) → 𝑥𝐶)
108, 9mpan2 703 . . . . 5 (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥𝐶)
11103ad2ant2 1076 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐶)
12 simp3 1056 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ≠ ∅)
13 mreintcl 16078 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥𝐶𝑥 ≠ ∅) → 𝑥𝐶)
147, 11, 12, 13syl3anc 1318 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐶)
15 sstr 3576 . . . . . . . 8 ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
161, 15mpan2 703 . . . . . . 7 (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
17163ad2ant2 1076 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝒫 𝐴)
18 intssuni2 4437 . . . . . 6 ((𝑥 ⊆ 𝒫 𝐴𝑥 ≠ ∅) → 𝑥 𝒫 𝐴)
1917, 12, 18syl2anc 691 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 𝒫 𝐴)
20 unipw 4845 . . . . 5 𝒫 𝐴 = 𝐴
2119, 20syl6sseq 3614 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐴)
22 elpw2g 4754 . . . . . 6 (𝐴𝐶 → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
2322adantl 481 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
24233ad2ant1 1075 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
2521, 24mpbird 246 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ 𝒫 𝐴)
2614, 25elind 3760 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (𝐶 ∩ 𝒫 𝐴))
272, 6, 26ismred 16085 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977   ≠ wne 2780   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  ∩ cint 4410  ‘cfv 5804  Moorecmre 16065 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-mre 16069 This theorem is referenced by:  submrc  16111
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