Theorem submre 17515

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) ∧ 𝐴 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))

Proof of Theorem submre

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 4187	. . 3 ⊢ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
2	1	a1i 11	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴)
3		simpr 484	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐶)
4		pwidg 4571	. . . 4 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ 𝒫 𝐴)
5	4	adantl 481	. . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝒫 𝐴)
6	3, 5	elind 4149	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ (𝐶 ∩ 𝒫 𝐴))
7		simp1l 1198	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝐶 ∈ (Moore‘𝑋))
8		inss1 4186	. . . . . 6 ⊢ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶
9		sstr 3939	. . . . . 6 ⊢ ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝐶) → 𝑥 ⊆ 𝐶)
10	8, 9	mpan2 691	. . . . 5 ⊢ (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝐶)
11	10	3ad2ant2 1134	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝐶)
12		simp3 1138	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ≠ ∅)
13		mreintcl 17505	. . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ 𝐶)
14	7, 11, 12, 13	syl3anc 1373	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ 𝐶)
15		sstr 3939	. . . . . . . 8 ⊢ ((𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ (𝐶 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
16	1, 15	mpan2 691	. . . . . . 7 ⊢ (𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) → 𝑥 ⊆ 𝒫 𝐴)
17	16	3ad2ant2 1134	. . . . . 6 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝒫 𝐴)
18		intssuni2 4925	. . . . . 6 ⊢ ((𝑥 ⊆ 𝒫 𝐴 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ ∪ 𝒫 𝐴)
19	17, 12, 18	syl2anc 584	. . . . 5 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ ∪ 𝒫 𝐴)
20		unipw 5395	. . . . 5 ⊢ ∪ 𝒫 𝐴 = 𝐴
21	19, 20	sseqtrdi 3971	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ 𝐴)
22		elpw2g 5275	. . . . . 6 ⊢ (𝐴 ∈ 𝐶 → (∩ 𝑥 ∈ 𝒫 𝐴 ↔ ∩ 𝑥 ⊆ 𝐴))
23	22	adantl 481	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → (∩ 𝑥 ∈ 𝒫 𝐴 ↔ ∩ 𝑥 ⊆ 𝐴))
24	23	3ad2ant1 1133	. . . 4 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → (∩ 𝑥 ∈ 𝒫 𝐴 ↔ ∩ 𝑥 ⊆ 𝐴))
25	21, 24	mpbird 257	. . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ 𝒫 𝐴)
26	14, 25	elind 4149	. 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) ∧ 𝑥 ⊆ (𝐶 ∩ 𝒫 𝐴) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ (𝐶 ∩ 𝒫 𝐴))
27	2, 6, 26	ismred 17512	1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝐴 ∈ 𝐶) → (𝐶 ∩ 𝒫 𝐴) ∈ (Moore‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2113   ≠ wne 2929   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4551  ∪ cuni 4860  ∩ cint 4899  ‘cfv 6489  Moorecmre 17492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-mre 17496

This theorem is referenced by:  submrc  17542