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Theorem submre 16618
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 4058 . . 3 (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
21a1i 11 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
3 simpr 479 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴𝐶)
4 pwidg 4393 . . . 4 (𝐴𝐶𝐴 ∈ 𝒫 𝐴)
54adantl 475 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴 ∈ 𝒫 𝐴)
63, 5elind 4025 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴 ∈ (𝐶 ∩ 𝒫 𝐴))
7 simp1l 1260 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
8 inss1 4057 . . . . . 6 (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶
9 sstr 3835 . . . . . 6 ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶) → 𝑥𝐶)
108, 9mpan2 684 . . . . 5 (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥𝐶)
11103ad2ant2 1170 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐶)
12 simp3 1174 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ≠ ∅)
13 mreintcl 16608 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥𝐶𝑥 ≠ ∅) → 𝑥𝐶)
147, 11, 12, 13syl3anc 1496 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐶)
15 sstr 3835 . . . . . . . 8 ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
161, 15mpan2 684 . . . . . . 7 (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
17163ad2ant2 1170 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝒫 𝐴)
18 intssuni2 4722 . . . . . 6 ((𝑥 ⊆ 𝒫 𝐴𝑥 ≠ ∅) → 𝑥 𝒫 𝐴)
1917, 12, 18syl2anc 581 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 𝒫 𝐴)
20 unipw 5139 . . . . 5 𝒫 𝐴 = 𝐴
2119, 20syl6sseq 3876 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐴)
22 elpw2g 5049 . . . . . 6 (𝐴𝐶 → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
2322adantl 475 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
24233ad2ant1 1169 . . . 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 4025 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (𝐶 ∩ 𝒫 𝐴))
272, 6, 26ismred 16615 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113  wcel 2166  wne 2999  cin 3797  wss 3798  c0 4144  𝒫 cpw 4378   cuni 4658   cint 4697  cfv 6123  Moorecmre 16595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-mre 16599
This theorem is referenced by:  submrc  16641
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