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Theorem submre 16580
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 4029 . . 3 (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
21a1i 11 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
3 simpr 478 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴𝐶)
4 pwidg 4364 . . . 4 (𝐴𝐶𝐴 ∈ 𝒫 𝐴)
54adantl 474 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴 ∈ 𝒫 𝐴)
63, 5elind 3996 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴 ∈ (𝐶 ∩ 𝒫 𝐴))
7 simp1l 1255 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
8 inss1 4028 . . . . . 6 (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶
9 sstr 3806 . . . . . 6 ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶) → 𝑥𝐶)
108, 9mpan2 683 . . . . 5 (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥𝐶)
11103ad2ant2 1165 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐶)
12 simp3 1169 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ≠ ∅)
13 mreintcl 16570 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥𝐶𝑥 ≠ ∅) → 𝑥𝐶)
147, 11, 12, 13syl3anc 1491 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐶)
15 sstr 3806 . . . . . . . 8 ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
161, 15mpan2 683 . . . . . . 7 (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
17163ad2ant2 1165 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝒫 𝐴)
18 intssuni2 4692 . . . . . 6 ((𝑥 ⊆ 𝒫 𝐴𝑥 ≠ ∅) → 𝑥 𝒫 𝐴)
1917, 12, 18syl2anc 580 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 𝒫 𝐴)
20 unipw 5109 . . . . 5 𝒫 𝐴 = 𝐴
2119, 20syl6sseq 3847 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐴)
22 elpw2g 5019 . . . . . 6 (𝐴𝐶 → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
2322adantl 474 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
24233ad2ant1 1164 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
2521, 24mpbird 249 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ 𝒫 𝐴)
2614, 25elind 3996 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (𝐶 ∩ 𝒫 𝐴))
272, 6, 26ismred 16577 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108  wcel 2157  wne 2971  cin 3768  wss 3769  c0 4115  𝒫 cpw 4349   cuni 4628   cint 4667  cfv 6101  Moorecmre 16557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-int 4668  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-mre 16561
This theorem is referenced by:  submrc  16603
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