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Theorem submre 16205
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 3818 . . 3 (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
21a1i 11 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
3 simpr 477 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴𝐶)
4 pwidg 4151 . . . 4 (𝐴𝐶𝐴 ∈ 𝒫 𝐴)
54adantl 482 . . 3 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴 ∈ 𝒫 𝐴)
63, 5elind 3782 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → 𝐴 ∈ (𝐶 ∩ 𝒫 𝐴))
7 simp1l 1083 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
8 inss1 3817 . . . . . 6 (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶
9 sstr 3596 . . . . . 6 ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶) → 𝑥𝐶)
108, 9mpan2 706 . . . . 5 (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥𝐶)
11103ad2ant2 1081 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐶)
12 simp3 1061 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ≠ ∅)
13 mreintcl 16195 . . . 4 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥𝐶𝑥 ≠ ∅) → 𝑥𝐶)
147, 11, 12, 13syl3anc 1323 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐶)
15 sstr 3596 . . . . . . . 8 ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
161, 15mpan2 706 . . . . . . 7 (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
17163ad2ant2 1081 . . . . . 6 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝒫 𝐴)
18 intssuni2 4474 . . . . . 6 ((𝑥 ⊆ 𝒫 𝐴𝑥 ≠ ∅) → 𝑥 𝒫 𝐴)
1917, 12, 18syl2anc 692 . . . . 5 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 𝒫 𝐴)
20 unipw 4889 . . . . 5 𝒫 𝐴 = 𝐴
2119, 20syl6sseq 3636 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥𝐴)
22 elpw2g 4797 . . . . . 6 (𝐴𝐶 → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
2322adantl 482 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
24233ad2ant1 1080 . . . 4 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ( 𝑥 ∈ 𝒫 𝐴 𝑥𝐴))
2521, 24mpbird 247 . . 3 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ 𝒫 𝐴)
2614, 25elind 3782 . 2 (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ∈ (𝐶 ∩ 𝒫 𝐴))
272, 6, 26ismred 16202 1 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wcel 1987  wne 2790  cin 3559  wss 3560  c0 3897  𝒫 cpw 4136   cuni 4409   cint 4447  cfv 5857  Moorecmre 16182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-mre 16186
This theorem is referenced by:  submrc  16228
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